Alternated hypercubic honeycomb

<table><tbody><tr><td>An <i>alternated square tiling</i> or <a href="/facts/Checkerboard/yIQaQoaM">checkerboard</a> pattern. or </td><td>An expanded square tiling.</td></tr><tr><td>A partially filled <i>alternated cubic honeycomb</i> with tetrahedral and octahedral cells. or </td><td>A subsymmetry colored alternated cubic honeycomb.</td></tr></tbody></table>
<p>In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, the alternated hypercube honeycomb (or demicubic honeycomb) is a dimensional infinite series of <a href="/facts/Honeycomb_(geometry)/seyRBcel">honeycombs</a>, based on the <a href="/facts/Hypercube_honeycomb/MQxLbDeB">hypercube honeycomb</a> with an <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternation</a> operation. It is given a <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of <a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter group</a> 
  
    
      
        
          
            
              
                B
                ~
              
            
          
          
            n
            −
            1
          
        
      
    
    {\displaystyle {\tilde {B}}_{n-1}}
  
 for n ≥ 4. A lower symmetry form 
  
    
      
        
          
            
              
                D
                ~
              
            
          
          
            n
            −
            1
          
        
      
    
    {\displaystyle {\tilde {D}}_{n-1}}
  
 can be created by removing another mirror on an order-4 <a href="/facts/Peak_(geometry)/T2E6BTU4">peak</a>.
</p><p>The alternated hypercube facets become <a href="/facts/Demihypercube/ackHKsEa">demihypercubes</a>, and the deleted vertices create new <a href="/facts/Orthoplex/erk0byF6">orthoplex</a> facets. The <a href="/facts/Vertex_figure/3u2ySK0C">vertex figure</a> for honeycombs of this family are <a href="/facts/Rectification_(geometry)/PkoUdq53">rectified</a> orthoplexes.
</p><p>These are also named as hδn for an (n-1)-dimensional honeycomb.
</p>
<table><tbody><tr><th rowspan="3">hδn</th><th rowspan="3">Name</th><th rowspan="3"><a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläflisymbol</a></th><th colspan="3"><a href="/facts/Coxeter_notation/ArUw6goe">Symmetry family</a></th></tr><tr><th>                                                                        B                ~                                                          n            −            1                                {\displaystyle {\tilde {B}}_{n-1}}  [4,3n-4,31,1]</th><th>                                                                        D                ~                                                          n            −            1                                {\displaystyle {\tilde {D}}_{n-1}}  [31,1,3n-5,31,1]</th></tr><tr><th colspan="3"><a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagrams</a> by family</th></tr><tr><td>hδ2</td><td><i><a href="/facts/Apeirogon/r0V9hleb">Apeirogon</a></i></td><td>{∞}</td><td></td><td></td></tr><tr><td>hδ3</td><td><i>Alternated square tiling</i>(Same as {4,4})</td><td>h{4,4}=t1{4,4}t0,2{4,4}</td><td></td><td></td></tr><tr><td>hδ4</td><td><i><a href="/facts/Alternated_cubic_honeycomb/mX4CxSvp">Alternated cubic honeycomb</a></i></td><td>h{4,3,4}{31,1,4}</td><td></td><td></td></tr><tr><td>hδ5</td><td><i><a href="/facts/16-cell_tetracomb/lbktV8Nr">16-cell tetracomb</a></i>(Same as {3,3,4,3})</td><td>h{4,32,4}{31,1,3,4}{31,1,1,1}</td><td></td><td></td></tr><tr><td>hδ6</td><td><i><a href="/facts/5-demicube_honeycomb/eYp2lMB8">5-demicube honeycomb</a></i></td><td>h{4,33,4}{31,1,32,4}{31,1,3,31,1}</td><td></td><td></td></tr><tr><td>hδ7</td><td><i><a href="/facts/6-demicube_honeycomb/oxttB0n0">6-demicube honeycomb</a></i></td><td>h{4,34,4}{31,1,33,4}{31,1,32,31,1}</td><td></td><td></td></tr><tr><td>hδ8</td><td><i><a href="/facts/7-demicube_honeycomb/QYm4RCGK">7-demicube honeycomb</a></i></td><td>h{4,35,4}{31,1,34,4}{31,1,33,31,1}</td><td></td><td></td></tr><tr><td>hδ9</td><td><i><a href="/facts/8-demicube_honeycomb/baWsKzQc">8-demicube honeycomb</a></i></td><td>h{4,36,4}{31,1,35,4}{31,1,34,31,1}</td><td></td><td></td></tr><tr><td colspan="6"> </td></tr><tr><td>hδn</td><td><i>n-demicubic honeycomb</i></td><td>h{4,3n-3,4}{31,1,3n-4,4}{31,1,3n-5,31,1}</td><td colspan="2">...</td></tr></tbody></table>

<ul><li><a href="/facts/Coxeter/D2l5jWGG">Coxeter, H.S.M.</a> <i><a href="/facts/Regular_Polytopes_(book)/jomuRNha">Regular Polytopes</a></i>, (3rd edition, 1973), Dover edition, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-61480-8
<ol><li>pp. 122–123, 1973. (The lattice of hypercubes γn form the <i>cubic honeycombs</i>, δn+1)</li>
<li>pp. 154–156: Partial truncation or alternation, represented by <i>h</i> prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}</li>
<li>p. 296, Table II: Regular honeycombs, δn+1</li></ol></li>
<li>Kaleidoscopes: Selected Writings of <a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">H. S. M. Coxeter</a>, editied 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></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>hδn</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>

Alternated hypercubic honeycomb open-in-new

Alternated hypercubic honeycomb