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Stericantic tesseractic honeycomb
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<h2 id="alternate-names">Alternate names</h2> <ul><li>Prismatotruncated demitesseractic tetracomb (pithatit)</li> <li>Small prismatodemitesseractic tetracomb</li></ul> <h2 id="related-honeycombs">Related honeycombs</h2> <p>The [4,3,31,1], , <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a> generates 31 permutations of uniform tessellations, 23 with distinct symmetry and 4 with distinct geometry. There are two alternated forms: the alternations (19) and (24) have the same geometry as the <a href="/facts/16-cell_honeycomb/lbktV8Nr">16-cell honeycomb</a> and <a href="/facts/Snub_24-cell_honeycomb/bzFwdh7X">snub 24-cell honeycomb</a> respectively. </p> <table><tbody><tr><th colspan="5">B4 honeycombs</th></tr><tr><th><a href="/facts/Coxeter_notation/ArUw6goe">Extendedsymmetry</a></th><th>Extendeddiagram</th><th>Order</th><th>Honeycombs</th></tr><tr><th>[4,3,31,1]:</th><th></th><th>×1</th><td><p> <a href="/facts/Steric_tesseractic_honeycomb/bNrEVScg">5</a>, <a href="/facts/Steriruncic_tesseractic_honeycomb/5bgsHj6T">6</a>, 7, <a href="/facts/Steriruncicantic_tesseractic_honeycomb/m7NFvIKv">8</a></p></td></tr><tr><th><[4,3,31,1]>:↔[4,3,3,4]</th><th> ↔ </th><th>×2</th><td><p> <a href="/facts/Tesseractic_honeycomb/fpucRJJn">9</a>, <a href="/facts/Rectified_tesseractic_honeycomb/P266Jc7y">10</a>, <a href="/facts/Truncated_tesseractic_honeycomb/SfgPsTUc">11</a>, <a href="/facts/Cantellated_tesseractic_honeycomb/2bEApvAq">12</a>, <a href="/facts/Bitruncated_tesseractic_honeycomb/CVRrn5Fy">13</a>, <a href="/facts/Canitruncated_tesseractic_honeycomb/zrAeIGXF">14</a>, </p><p> <a href="/facts/Rectified_tesseractic_honeycomb/P266Jc7y">(10)</a>, <a href="/facts/Runcinated_tesseractic_honeycomb/7EpkHRm5">15</a>, <a href="/facts/Runcitruncated_tesseractic_honeycomb/yCK6fPDn">16</a>, <a href="/facts/Bitruncated_tesseractic_honeycomb/CVRrn5Fy">(13)</a>, <a href="/facts/Runcicantellated_tesseractic_honeycomb/beTR5ecb">17</a>, <a href="/facts/Runcicantitruncated_tesseractic_honeycomb/un2mgdGr">18</a>, <a href="/facts/16-cell_honeycomb/lbktV8Nr">19</a></p></td></tr><tr><th>[3[1+,4,3,31,1]]↔ [3[3,31,1,1]]↔ [3,3,4,3]</th><th>↔ ↔ </th><th>×3</th><td><p> <a href="/facts/16-cell_honeycomb/lbktV8Nr">1</a>, <a href="/facts/Truncated_16-cell_honeycomb/J82DsBNc">2</a>, <a href="/facts/Birectified_16-cell_honeycomb/5UBjSBSx">3</a>, <a href="/facts/Bitruncated_16-cell_honeycomb/mh2clyq5">4</a></p></td></tr><tr><th>[(3,3)[1+,4,3,31,1]]↔ [(3,3)[31,1,1,1]]↔ [3,4,3,3]</th><th>↔ ↔ </th><th>×12</th><td><p> <a href="/facts/24-cell_honeycomb/foFoK5YX">20</a>, <a href="/facts/Rectified_24-cell_honeycomb/8ab5I5u9">21</a>, <a href="/facts/Truncated_24-cell_honeycomb/uQN2kXg1">22</a>, <a href="/facts/Snub_24-cell_honeycomb/bzFwdh7X">23</a></p></td></tr></tbody></table> <h2 id="see-also">See also</h2> <p>Regular and uniform honeycombs in 4-space: </p> <ul><li><a href="/facts/Tesseractic_honeycomb/fpucRJJn">Tesseractic honeycomb</a></li> <li><a href="/facts/16-cell_honeycomb/lbktV8Nr">16-cell honeycomb</a></li> <li><a href="/facts/24-cell_honeycomb/foFoK5YX">24-cell honeycomb</a></li> <li><a href="/facts/Rectified_24-cell_honeycomb/8ab5I5u9">Rectified 24-cell honeycomb</a></li> <li><a href="/facts/Truncated_24-cell_honeycomb/uQN2kXg1">Truncated 24-cell honeycomb</a></li> <li><a href="/facts/Snub_24-cell_honeycomb/bzFwdh7X">Snub 24-cell honeycomb</a></li> <li><a href="/facts/5-cell_honeycomb/YgY6eSTr">5-cell honeycomb</a></li> <li><a href="/facts/Truncated_5-cell_honeycomb/YgY6eSTr">Truncated 5-cell honeycomb</a></li> <li><a href="/facts/Omnitruncated_5-cell_honeycomb/YgY6eSTr">Omnitruncated 5-cell honeycomb</a></li></ul> <h2 id="notes">Notes</h2> <ul><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 24) H.S.M. Coxeter, <i>Regular and Semi-Regular Polytopes III</i>, [Math. Zeit. 200 (1988) 3-45]</li></ul></li> <li>George Olshevsky, <i>Uniform Panoploid Tetracombs</i>, Manuscript (2006) <i>(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)</i></li> <li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/flat.htm">"4D Euclidean tesselations"</a>. x3x3o *b3o4x - pithatit - O109</li></ul> <table><tbody><tr><th colspan="7"><ul><li>v</li><li>t</li><li>e</li></ul>Fundamental convex <a href="/facts/Regular_honeycomb/Kng25Ymq">regular</a> and <a href="/facts/Uniform_honeycomb/fGXtrgGF">uniform honeycombs</a> in dimensions 2–9</th></tr><tr><th>Space</th><th><a href="/facts/Coxeter_group/bgzHFm1Q">Family</a></th><td> A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} </td><td> C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} </td><td> B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} </td><td> D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} </td><td> G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}} </td></tr><tr><th>E2</th><th><a href="/facts/Uniform_tiling/QVK02ksI">Uniform tiling</a></th><td><a href="/facts/Triangular_tiling/Jx8f8Krh">0[3]</a></td><td><a href="/facts/Square_tiling/3ym86mvv">δ3</a></td><td><a href="/facts/Square_tiling/3ym86mvv">hδ3</a></td><td><a href="/facts/Square_tiling/3ym86mvv">qδ3</a></td><td><a href="/facts/Hexagonal_tiling/HwG5F0YH">Hexagonal</a></td></tr><tr><th>E3</th><th><a href="/facts/Uniform_convex_honeycomb/78IkTl7A">Uniform convex honeycomb</a></th><td><a href="/facts/Tetrahedral-octahedral_honeycomb/mX4CxSvp">0[4]</a></td><td><a href="/facts/Cubic_honeycomb/eMxySFur">δ4</a></td><td><a href="/facts/Tetrahedral-octahedral_honeycomb/mX4CxSvp">hδ4</a></td><td><a href="/facts/Quarter_cubic_honeycomb/22qQazbw">qδ4</a></td><td></td></tr><tr><th>E4</th><th><a href="/facts/Uniform_4-honeycomb/mqxsoqwY">Uniform 4-honeycomb</a></th><td><a href="/facts/5-cell_honeycomb/YgY6eSTr">0[5]</a></td><td><a href="/facts/Tesseractic_honeycomb/fpucRJJn">δ5</a></td><td><a href="/facts/16-cell_honeycomb/lbktV8Nr">hδ5</a></td><td><a href="/facts/Rectified_tesseractic_honeycomb/P266Jc7y">qδ5</a></td><td><a href="/facts/24-cell_honeycomb/foFoK5YX">24-cell honeycomb</a></td></tr><tr><th>E5</th><th><a href="/facts/Uniform_5-honeycomb/pxqFTw2F">Uniform 5-honeycomb</a></th><td><a href="/facts/5-simplex_honeycomb/XSLrRsLz">0[6]</a></td><td><a href="/facts/5-cubic_honeycomb/bqAJvIBI">δ6</a></td><td><a href="/facts/5-demicubic_honeycomb/eYp2lMB8">hδ6</a></td><td><a href="/facts/Quarter_5-cubic_honeycomb/r3AAQHbg">qδ6</a></td><td></td></tr><tr><th>E6</th><th><a href="/facts/Uniform_6-honeycomb/suCJSvTN">Uniform 6-honeycomb</a></th><td><a href="/facts/6-simplex_honeycomb/H7LWND7E">0[7]</a></td><td><a href="/facts/6-cubic_honeycomb/2FdezLvD">δ7</a></td><td><a href="/facts/6-demicubic_honeycomb/oxttB0n0">hδ7</a></td><td><a href="/facts/Quarter_6-cubic_honeycomb/vDRQk9kS">qδ7</a></td><td><a href="/facts/2_22_honeycomb/cvXxbk6L">222</a></td></tr><tr><th>E7</th><th><a href="/facts/Uniform_7-honeycomb/M93v3h3c">Uniform 7-honeycomb</a></th><td><a href="/facts/7-simplex_honeycomb/dq8hy2lI">0[8]</a></td><td><a href="/facts/7-cubic_honeycomb/Im2bVBf7">δ8</a></td><td><a href="/facts/7-demicubic_honeycomb/QYm4RCGK">hδ8</a></td><td><a href="/facts/Quarter_7-cubic_honeycomb/GjVHMBLq">qδ8</a></td><td><a href="/facts/1_33_honeycomb/tYL93yF1">133</a> • <a href="/facts/3_31_honeycomb/VlInrcea">331</a></td></tr><tr><th>E8</th><th><a href="/facts/Uniform_8-honeycomb/UdFL9ns2">Uniform 8-honeycomb</a></th><td><a href="/facts/8-simplex_honeycomb/eoAGHg7J">0[9]</a></td><td><a href="/facts/8-cubic_honeycomb/fx3hM5DD">δ9</a></td><td><a href="/facts/8-demicubic_honeycomb/baWsKzQc">hδ9</a></td><td>qδ9</td><td><a href="/facts/1_52_honeycomb/LAJXFvnV">152</a> • <a href="/facts/2_51_honeycomb/XthKHy0m">251</a> • <a href="/facts/5_21_honeycomb/f971VRma">521</a></td></tr><tr><th>E9</th><th><a href="/facts/Uniform_9-honeycomb/Qc5NglV8">Uniform 9-honeycomb</a></th><td>0[10]</td><td>δ10</td><td>hδ10</td><td>qδ10</td><td></td></tr><tr><th>E10</th><th>Uniform 10-honeycomb</th><td>0[11]</td><td>δ11</td><td>hδ11</td><td>qδ11</td><td></td></tr><tr><th>E<i>n</i>−1</th><th>Uniform (<i>n</i>−1)-<a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycomb</a></th><td><a href="/facts/Simplectic_honeycomb/hCmbARWt">0[<i>n</i>]</a></td><td><a href="/facts/Hypercubic_honeycomb/MQxLbDeB">δ<i>n</i></a></td><td><a href="/facts/Alternated_hypercubic_honeycomb/zrU9sG2e">hδ<i>n</i></a></td><td><a href="/facts/Quarter_hypercubic_honeycomb/7Je1f2st">qδ<i>n</i></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></tr></tbody></table>