8-cubic honeycomb
(no image)
Type	Regular 8-honeycomb Uniform 8-honeycomb
Family	Hypercube honeycomb
Schläfli symbol	{4,3⁶,4} {4,3⁵,3^1,1} t_0,8{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

In geometry, 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, , contains all trirectified 8-orthoplex facets and is the Voronoi tessellation of the D₈^* lattice. Facets can be identically colored from a doubled ${\displaystyle {\tilde {C))_{8))$ ×2, [[4,3⁶,4]] symmetry, alternately colored from ${\displaystyle {\tilde {C))_{8))$ , [4,3⁶,4] symmetry, three colors from ${\displaystyle {\tilde {B))_{8))$ , [4,3⁵,3^1,1] symmetry, and 4 colors from ${\displaystyle {\tilde {D))_{8))$ , [3^1,1,3⁴,3^1,1] symmetry.

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]

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A))_{n-1))$	${\displaystyle {\tilde {C))_{n-1))$	${\displaystyle {\tilde {B))_{n-1))$	${\displaystyle {\tilde {D))_{n-1))$	${\displaystyle {\tilde {G))_{2))$ / ${\displaystyle {\tilde {F))_{4))$ / ${\displaystyle {\tilde {E))_{n-1))$
E²	Uniform tiling	{3^[3]}	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	{3^[4]}	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	{3^[5]}	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	{3^[6]}	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	{3^[7]}	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	{3^[8]}	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	{3^[9]}	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	{3^[10]}	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	{3^[11]}	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	{3^[n]}	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

Related honeycombs

Quadrirectified 8-cubic honeycomb

See also

References