6-cubic honeycomb | |
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(no image) | |
Type | Regular 6-honeycomb Uniform 6-honeycomb |
Family | Hypercube honeycomb |
Schläfli symbol | {4,34,4} {4,33,31,1} |
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 | , [4,34,4] , [4,33,31,1] |
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.
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).
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.
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.