8-cubic honeycomb

8-cubic honeycomb
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Type	Regular 8-dimensional honeycomb
Family	Hypercube honeycomb
Schläfli symbol	{4,36,4} {4,35,31,1} t0,8{4,36,4} {∞}8
Coxeter-Dynkin diagrams
8-face type	{4,36}
7-face type	{4,35}
6-face type	{4,34}
5-face type	{4,33}
4-face type	{4,32}
Cell type	{4,3}
Face type	{4}
Face figure	{4,3} (octahedron)
Edge figure	8 {4,3,3} (16-cell)
Vertex figure	256 {4,36} (8-orthoplex)
Coxeter group	[4,36,4]
Dual	self-dual
Properties	vertex-transitive, edge-transitive, face-transitive, cell-transitive

The 8-cubic honeycomb or octeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 8-space.

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space, and the tesseractic honeycomb of 4-space.

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,3⁶,4}. Another form has two alternating hypercube facets (like a checkerboard) with Schläfli symbol {4,3⁵,3^1,1}. The lowest symmetry Wythoff construction has 256 types of facets around each vertex and a prismatic product Schläfli symbol {∞}⁸.

Related honeycombs

The [4,3⁶,4], , Coxeter group generates 511 permutations of uniform tessellations, 271 with unique symmetry and 270 with unique geometry. The expanded 8-cubic honeycomb is geometrically identical to the 8-cubic honeycomb.

The 8-cubic honeycomb can be alternated into the 8-demicubic honeycomb, replacing the 8-cubes with 8-demicubes, and the alternated gaps are filled by 8-orthoplex facets.

Quadrirectified 8-cubic honeycomb

A quadrirectified 8-cubic honeycomb, , containins all trirectified 8-orthoplex facets and is the Voronoi tessellation of the D₈^* lattice. Facets can be identically colored from a doubled ${\tilde{C}}_8$×2, [[4,3⁶,4]] symmetry, alternately colored from ${\tilde{C}}_8$, [4,3⁶,4] symmetry, three colors from ${\tilde{B}}_8$, [4,3⁵,3^1,1] symmetry, and 4 colors from ${\tilde{D}}_8$, [3^1,1,3⁴,3^1,1] symmetry.

See also

• List of regular polytopes

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• 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, 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]