Hypercubic honeycomb

File:Square tiling uniform coloring 1.svg A regular square tiling. Template:CDD 1 color	File:Partial cubic honeycomb.png A cubic honeycomb in its regular form. Template:CDD 1 color
File:Square tiling uniform coloring 7.png A checkboard square tiling Template:CDD 2 colors	File:Bicolor cubic honeycomb.png A cubic honeycomb checkerboard. Template:CDD 2 colors
File:Square tiling uniform coloring 8.png Expanded square tiling Template:CDD 3 colors	File:Runcinated cubic honeycomb.png Expanded cubic honeycomb Template:CDD 4 colors
File:Square tiling uniform coloring 9.png Template:CDD 4 colors	File:Cubic 8-color honeycomb.png Template:CDD 8 colors

In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in Template:Mvar-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 Template:Mvar-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 Template:Mvar-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, Template:Mvar colors)	Checkerboard {4,3n−4,31,1} (2 colors)
δ2	Apeirogon	{∞}	Template:CDD
δ3	Square tiling	{∞}(2) {4,4}	Template:CDD	Template:CDD Template:CDD	Template:CDD
δ4	Cubic honeycomb	{∞}(3) {4,3,4} {4,31,1}	Template:CDD	Template:CDD Template:CDD	Template:CDD
δ5	4-cube honeycomb	{∞}(4) {4,32,4} {4,3,31,1}	Template:CDD	Template:CDD Template:CDD	Template:CDD
δ6	5-cube honeycomb	{∞}(5) {4,33,4} {4,32,31,1}	Template:CDD	Template:CDD Template:CDD	Template:CDD
δ7	6-cube honeycomb	{∞}(6) {4,34,4} {4,33,31,1}	Template:CDD	Template:CDD Template:CDD	Template:CDD
δ8	7-cube honeycomb	{∞}(7) {4,35,4} {4,34,31,1}	Template:CDD	Template:CDD Template:CDD	Template:CDD
δ9	8-cube honeycomb	{∞}(8) {4,36,4} {4,35,31,1}	Template:CDD	Template:CDD Template:CDD	Template:CDD
δn	Template:Mvar-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, Template:ISBN
  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