6-cubic honeycomb

6-cubic honeycomb(no image)TypeRegular 6-honeycombUniform 6-honeycombFamilyHypercube honeycombSchläfli symbol {4,34,4}{4,33,31,1}Coxeter-Dynkin diagrams6-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 figure8 {4,3,3}(16-cell)Vertex figure64 {4,34}(6-orthoplex)Coxeter group, [4,34,4], [4,33,31,1]Dualself-dualPropertiesvertex-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], , Coxeter group generates 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The 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, , contains all birectified 6-orthoplex facets and is the Voronoi tessellation of the D6* lattice. Facets can be identically colored from a doubled ×2, [4,34,4] symmetry, alternately colored from , [4,34,4] symmetry, three colors from , [4,33,31,1] symmetry, and 4 colors from , [31,1,3,3,31,1] symmetry.

6-cubic honeycomb

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