A regular square tiling. 1 color	A cubic honeycomb in its regular form. 1 color
A checkboard square tiling 2 colors	A cubic honeycomb checkerboard. 2 colors
Expanded square tiling 3 colors	Expanded cubic honeycomb 4 colors
4 colors	8 colors

In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in $n$ -dimensional spaces with the Schläfli symbols ${4,3...3,4}$ and containing the symmetry of Coxeter group $R n$ (or $B ~ n -1$ ) for $n \geq 3$ .

The tessellation is constructed from 4 $n$ -hypercubes per ridge. The vertex figure is a cross-polytope ${3...3,4}.$

The hypercubic honeycombs are self-dual.

Coxeter named this family as $δ n+1$ for an $n$ -dimensional honeycomb.

Wythoff construction classes by dimension

A Wythoff construction is a method for constructing a uniform polyhedron or plane tiling.

The two general forms of the hypercube honeycombs are the regular form with identical hypercubic facets and one semiregular, with alternating hypercube facets, like a checkerboard.

A third form is generated by an expansion operation applied to the regular form, creating facets in place of all lower-dimensional elements. For example, an expanded cubic honeycomb has cubic cells centered on the original cubes, on the original faces, on the original edges, on the original vertices, creating 4 colors of cells around in vertex in 1:3:3:1 counts.

The orthotopic honeycombs are a family topologically equivalent to the cubic honeycombs but with lower symmetry, in which each of the three axial directions may have different edge lengths. The facets are hyperrectangles, also called orthotopes; in 2 and 3 dimensions the orthotopes are rectangles and cuboids respectively.

$δ n$	Name	Schläfli symbols	Coxeter-Dynkin diagrams
$δ n$	Name	Schläfli symbols	Orthotopic ${\infty} (n) (2 m$ colors, $m < n)$	Regular (Expanded) ${4,3 n -1,4}$ (1 color, $n$ colors)	Checkerboard ${4,3 n -4,3 1,1}$ (2 colors)
$δ 2$	Apeirogon	${\infty}$
$δ 3$	Square tiling	${\infty} (2) {4,4}$
$δ 4$	Cubic honeycomb	${\infty} (3) {4,3,4} {4,3 1,1}$
$δ 5$	4-cube honeycomb	${\infty} (4) {4,3 2,4} {4,3,3 1,1}$
$δ 6$	5-cube honeycomb	${\infty} (5) {4,3 3,4} {4,3 2,3 1,1}$
$δ 7$	6-cube honeycomb	${\infty} (6) {4,3 4,4} {4,3 3,3 1,1}$
$δ 8$	7-cube honeycomb	${\infty} (7) {4,3 5,4} {4,3 4,3 1,1}$
$δ 9$	8-cube honeycomb	${\infty} (8) {4,3 6,4} {4,3 5,3 1,1}$
$δ n$	$n$ -hypercubic honeycomb	${\infty} (n) {4,3 n-3,4} {4,3 n-4,3 1,1}$	...

References

Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
1. pp. 122–123. (The lattice of hypercubes γ_n form the cubic honeycombs, δ_n+1)
2. pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={3^1,1,4}, h{4,3,3,4}={3,3,4,3}
3. p. 296, Table II: Regular honeycombs, δ_n+1

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

Wythoff construction classes by dimension

See also

References