6-cubic honeycomb
(no image)
Type
Regular 6-honeycomb Uniform 6-honeycomb
Family
Hypercube honeycomb
Schläfli symbol {4,34 ,4}3 ,31,1 }
Coxeter-Dynkin diagrams
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6-face type
{4,34 }
5-face type
{4,33 }
4-face type
{4,3,3}
Cell type
{4,3}
Face type
{4}
Face figure
{4,3} octahedron )
Edge figure
8 {4,3,3} 16-cell )
Vertex figure
64 {4,34 } 6-orthoplex )
Coxeter group C ~ 6 4 ,4]B ~ 6 3 ,31,1 ]
Dual
self-dual
Properties
vertex-transitive , edge-transitive , face-transitive , cell-transitive
The 6-cubic honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb ) in Euclidean 6-space.
It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.
Constructions There are many different Wythoff constructions of this honeycomb . The most symmetric form is regular , with Schläfli symbol {4,34 ,4}. Another form has two alternating 6-cube facets (like a checkerboard ) with Schläfli symbol {4,33 ,31,1 }. The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol {∞}(6) .
Related honeycombs The [4,34 ,4], File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb.
The 6-cubic honeycomb can be alternated into the 6-demicubic honeycomb , replacing the 6-cubes with 6-demicubes , and the alternated gaps are filled by 6-orthoplex facets.
Trirectified 6-cubic honeycomb A trirectified 6-cubic honeycomb , File:CDel node 1.png File:CDel split1.png File:CDel nodes.png File:CDel 3ab.png File:CDel nodes.png File:CDel 4a4b.png File:CDel nodes.png birectified 6-orthoplex facets and is the Voronoi tessellation of the D6 * lattice . Facets can be identically colored from a doubled C ~ 6 4 ,4] symmetry, alternately colored from C ~ 6 4 ,4] symmetry, three colors from B ~ 6 3 ,31,1 ] symmetry, and 4 colors from D ~ 6 1,1 ,3,3,31,1 ] symmetry.
See also References Coxeter, H.S.M. Regular Polytopes ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombsKaleidoscopes: Selected Writings of H. S. M. Coxeter , edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III , [Math. Zeit. 200 (1988) 3-45] 
