Rectified 8-orthoplexes

<h2 id="rectified-8-orthoplex">Rectified 8-orthoplex</h2>
<table><tbody><tr><th colspan="2">Rectified 8-orthoplex</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>t1{3,3,3,3,3,3,4}</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>272</td></tr><tr><td>6-faces</td><td>3072</td></tr><tr><td>5-faces</td><td>8960</td></tr><tr><td>4-faces</td><td>12544</td></tr><tr><td>Cells</td><td>10080</td></tr><tr><td>Faces</td><td>4928</td></tr><tr><td>Edges</td><td>1344</td></tr><tr><td>Vertices</td><td>112</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td>6-orthoplex prism</td></tr><tr><td><a href="/facts/Petrie_polygon/uDmot1hD">Petrie polygon</a></td><td><a href="/facts/Hexakaidecagon/3jTUD0zD">hexakaidecagon</a></td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a></td><td>C8, [4,36]D8, [35,1,1]</td></tr><tr><td>Properties</td><td><a href="/facts/Convex_polytope/2pRHcRgC">convex</a></td></tr></tbody></table>
<p>The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the <a href="/facts/Simple_Lie_group/LkHTNYU9">simple Lie group</a> D8. The vertices can be seen in 3 <a href="/facts/Hyperplane/2s2ycYYd">hyperplanes</a>, with the 28 vertices <a href="/facts/Rectified_7-simplex/kwsH3Rdg">rectified 7-simplexs</a> cells on opposite sides, and 56 vertices of an <a href="/facts/Expanded_7-simplex/1CTlxcXG">expanded 7-simplex</a> passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a>.
</p>
<h3>Related polytopes</h3>
<p>The <i>rectified 8-orthoplex</i> is the <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> for the <a href="/facts/Demiocteractic_honeycomb/baWsKzQc">demiocteractic honeycomb</a>.
</p>
 or 
<h3>Alternate names</h3>
<ul><li>rectified octacross</li>
<li>rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a></li></ul>
<h3>Construction</h3>
<p>There are two <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a> associated with the <i>rectified 8-orthoplex</i>, one with the C8 or [4,36] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or [35,1,1] Coxeter group.
</p>
<h3>Cartesian coordinates</h3>
<p><a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> for the vertices of a rectified 8-orthoplex, centered at the origin, edge length 
  
    
      
        
          
            2
          
        
      
    
    {\displaystyle {\sqrt {2}}}
  
 are all permutations of:
</p>
(±1,±1,0,0,0,0,0,0)
<h3>Images</h3>
<a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projections</a><table><tbody><tr><th colspan="3">B8</th><th colspan="3">B7</th></tr><tr><td colspan="3"></td><td colspan="3"></td></tr><tr><td colspan="3">[16]</td><td colspan="3">[14]</td></tr><tr><th colspan="3">B6</th><th colspan="3">B5</th></tr><tr><td colspan="3"></td><td colspan="3"></td></tr><tr><td colspan="3">[12]</td><td colspan="3">[10]</td></tr><tr><th colspan="2">B4</th><th colspan="2">B3</th><th colspan="2">B2</th></tr><tr><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td></tr><tr><td colspan="2">[8]</td><td colspan="2">[6]</td><td colspan="2">[4]</td></tr><tr><th colspan="2">A7</th><th colspan="2">A5</th><th colspan="2">A3</th></tr><tr><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td></tr><tr><td colspan="2">[8]</td><td colspan="2">[6]</td><td colspan="2">[4]</td></tr></tbody></table>
<h2 id="birectified-8-orthoplex">Birectified 8-orthoplex</h2>
<table><tbody><tr><th colspan="2">Birectified 8-orthoplex</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>t2{3,3,3,3,3,3,4}</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>272</td></tr><tr><td>6-faces</td><td>3184</td></tr><tr><td>5-faces</td><td>16128</td></tr><tr><td>4-faces</td><td>34048</td></tr><tr><td>Cells</td><td>36960</td></tr><tr><td>Faces</td><td>22400</td></tr><tr><td>Edges</td><td>6720</td></tr><tr><td>Vertices</td><td>448</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td>{3,3,3,4}x{3}</td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a></td><td>C8, [3,3,3,3,3,3,4]D8, [35,1,1]</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>birectified octacross</li>
<li>birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a></li></ul>
<h3>Cartesian coordinates</h3>
<p><a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> for the vertices of a birectified 8-orthoplex, centered at the origin, edge length 
  
    
      
        
          
            2
          
        
      
    
    {\displaystyle {\sqrt {2}}}
  
 are all permutations of:
</p>
(±1,±1,±1,0,0,0,0,0)
<h3>Images</h3>
<a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projections</a><table><tbody><tr><th colspan="3">B8</th><th colspan="3">B7</th></tr><tr><td colspan="3"></td><td colspan="3"></td></tr><tr><td colspan="3">[16]</td><td colspan="3">[14]</td></tr><tr><th colspan="3">B6</th><th colspan="3">B5</th></tr><tr><td colspan="3"></td><td colspan="3"></td></tr><tr><td colspan="3">[12]</td><td colspan="3">[10]</td></tr><tr><th colspan="2">B4</th><th colspan="2">B3</th><th colspan="2">B2</th></tr><tr><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td></tr><tr><td colspan="2">[8]</td><td colspan="2">[6]</td><td colspan="2">[4]</td></tr><tr><th colspan="2">A7</th><th colspan="2">A5</th><th colspan="2">A3</th></tr><tr><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td></tr><tr><td colspan="2">[8]</td><td colspan="2">[6]</td><td colspan="2">[4]</td></tr></tbody></table>
<h2 id="trirectified-8-orthoplex">Trirectified 8-orthoplex</h2>
<table><tbody><tr><th colspan="2">Trirectified 8-orthoplex</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>t3{3,3,3,3,3,3,4}</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>16+256</td></tr><tr><td>6-faces</td><td>1024 + 2048 + 112</td></tr><tr><td>5-faces</td><td>1792 + 7168 + 7168 + 448</td></tr><tr><td>4-faces</td><td>1792 + 10752 + 21504 + 14336</td></tr><tr><td>Cells</td><td>8960 + 126880 + 35840</td></tr><tr><td>Faces</td><td>17920 + 35840</td></tr><tr><td>Edges</td><td>17920</td></tr><tr><td>Vertices</td><td>1120</td></tr><tr><td><a href="/facts/Vertex_figure/3u2ySK0C">Vertex figure</a></td><td>{3,3,4}x{3,3}</td></tr><tr><td><a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter groups</a></td><td>C8, [3,3,3,3,3,3,4]D8, [35,1,1]</td></tr><tr><td>Properties</td><td><a href="/facts/Convex_polytope/2pRHcRgC">convex</a></td></tr></tbody></table>
<p>The trirectified 8-orthoplex can <a href="/facts/Tessellation/7BWDywXC">tessellate</a> space in the <a href="/facts/Quadrirectified_8-cubic_honeycomb/fx3hM5DD">quadrirectified 8-cubic honeycomb</a>.
</p>
<h3>Alternate names</h3>
<ul><li>trirectified octacross</li>
<li>trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li></ul>
<h3>Cartesian coordinates</h3>
<p><a href="/facts/Cartesian_coordinates/lkTqvMmB">Cartesian coordinates</a> for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length 
  
    
      
        
          
            2
          
        
      
    
    {\displaystyle {\sqrt {2}}}
  
 are all permutations of:
</p>
(±1,±1,±1,±1,0,0,0,0)
<h3>Images</h3>
<a href="/facts/Orthographic_projection/L3Cv9kYD">orthographic projections</a><table><tbody><tr><th colspan="3">B8</th><th colspan="3">B7</th></tr><tr><td colspan="3"></td><td colspan="3"></td></tr><tr><td colspan="3">[16]</td><td colspan="3">[14]</td></tr><tr><th colspan="3">B6</th><th colspan="3">B5</th></tr><tr><td colspan="3"></td><td colspan="3"></td></tr><tr><td colspan="3">[12]</td><td colspan="3">[10]</td></tr><tr><th colspan="2">B4</th><th colspan="2">B3</th><th colspan="2">B2</th></tr><tr><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td></tr><tr><td colspan="2">[8]</td><td colspan="2">[6]</td><td colspan="2">[4]</td></tr><tr><th colspan="2">A7</th><th colspan="2">A5</th><th colspan="2">A3</th></tr><tr><td colspan="2"></td><td colspan="2"></td><td colspan="2"></td></tr><tr><td colspan="2">[8]</td><td colspan="2">[6]</td><td colspan="2">[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/polyzetta.htm">"8D uniform polytopes (polyzetta)"</a>. o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark</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>Klitzing, (o3x3o3o3o3o3o4o - rek) <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Klitzing, (o3o3x3o3o3o3o4o - bark) <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Klitzing, (o3o3o3x3o3o3o4o - tark) <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Rectified 8-orthoplexes open-in-new

Rectified 8-orthoplexes