Hypercubic honeycomb

Short description: Regular tilings of ≥3D spaces with hypercubes

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 ≥ 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.

δn	Name	Schläfli symbols	Coxeter-Dynkin diagrams
			Orthotopic {∞}(n) (2m colors, m < n)	Regular (Expanded) {4,3n–1,4} (1 color, n colors)	Checkerboard {4,3n–4,31,1} (2 colors)
δ2	Apeirogon	{∞}
δ3	Square tiling	{∞}(2) {4,4}
δ4	Cubic honeycomb	{∞}(3) {4,3,4} {4,31,1}
δ5	4-cube honeycomb	{∞}(4) {4,32,4} {4,3,31,1}
δ6	5-cube honeycomb	{∞}(5) {4,33,4} {4,32,31,1}
δ7	6-cube honeycomb	{∞}(6) {4,34,4} {4,33,31,1}
δ8	7-cube honeycomb	{∞}(7) {4,35,4} {4,34,31,1}
δ9	8-cube honeycomb	{∞}(8) {4,36,4} {4,35,31,1}
δn	n-hypercubic honeycomb	{∞}(n) {4,3n-3,4} {4,3n-4,31,1}	...

See also

• Alternated hypercubic honeycomb
• Quarter hypercubic honeycomb
• Simplectic honeycomb
• Truncated simplectic honeycomb
• Omnitruncated simplectic honeycomb

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

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	[math]\displaystyle{ {\tilde{A}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{C}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{B}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{D}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
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
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21