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8-cubic honeycomb
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Type	Regular 8-honeycomb Uniform 8-honeycomb
Family	Hypercube honeycomb
Schläfli symbol	{4,3,4} {4,3,3} t0,8{4,3,4} {∞}
Coxeter-Dynkin diagrams
8-face type	{4,3}
7-face type	{4,3}
6-face type	{4,3}
5-face type	{4,3}
4-face type	{4,3}
Cell type	{4,3}
Face type	{4}
Face figure	{4,3} (octahedron)
Edge figure	8 {4,3,3} (16-cell)
Vertex figure	256 {4,3} (8-orthoplex)
Coxeter group
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 {4,3,4}. Another form has two alternating hypercube facets (like a checkerboard) with Schläfli symbol {4,3,3}. The lowest symmetry Wythoff construction has 256 types of facets around each vertex and a prismatic product Schläfli symbol {∞}.

Related honeycombs

The , , 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, ] symmetry, alternately colored from ${\displaystyle {\tilde {C}}_{8}}$, symmetry, three colors from ${\displaystyle {\tilde {B}}_{8}}$, symmetry, and 4 colors from ${\displaystyle {\tilde {D}}_{8}}$, 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
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E	Uniform tiling	0	δ3	hδ3	qδ3	Hexagonal
E	Uniform convex honeycomb	0	δ4	hδ4	qδ4
E	Uniform 4-honeycomb	0	δ5	hδ5	qδ5	24-cell honeycomb
E	Uniform 5-honeycomb	0	δ6	hδ6	qδ6
E	Uniform 6-honeycomb	0	δ7	hδ7	qδ7	222
E	Uniform 7-honeycomb	0	δ8	hδ8	qδ8	133 • 331
E	Uniform 8-honeycomb	0	δ9	hδ9	qδ9	152 • 251 • 521
E	Uniform 9-honeycomb	0	δ10	hδ10	qδ10
E	Uniform 10-honeycomb	0	δ11	hδ11	qδ11
E	Uniform (n-1)-honeycomb	0	δn	hδn	qδn	1k2 • 2k1 • k21

• Honeycombs (geometry)
• 9-polytopes
• Regular tessellations