Hypercubic honeycomb

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A regular square tiling.

A partial-filled cubic honeycomb in its regular form.

A partially-filled cubic honeycomb in its semiregular form.

In geometry, a hypercubic honeycomb is a family of regular honeycombs (tessellations) in n-dimensions 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}.

These are also named as - δ_n+1 for an n-dimensional honeycomb.

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

A more general class of honeycombs are called, orthotopic honeycombs, with identical topology, but allow each axial direction to have different edge lengths, for example with rectangle and cuboid facets in 2 and 3 dimensions.

δn	Name	Schläfli symbol	Coxeter-Dynkin diagrams
			Orthotopic	Regular	Semiregular
δ2	Apeirogon	{∞}
δ3	Square tiling	{4,4}
δ4	Cubic honeycomb	{4,3,4}
δ5	Tesseractic tetracomb	{4,32,4}
δ6	Penteractic pentacomb	{4,33,4}
δ7	Hexeractic hexacomb	{4,34,4}
δ8	Hepteractic heptacomb	{4,35,4}
δ9	Octeractic octacomb	{4,36,4}
δ10	Enneractic enneacomb	{4,37,4}	...

[edit] Alternated hypercubic honeycombs

An alternated square tiling is another square tiling, but having two types of squares, alternating in a checkerboard pattern.	A twice alternated square tiling.
A partially-filled alternated cubic honeycomb with tetrahedral and octahedral cells.	A subsymmetry colored alternated cubic honeycomb.

A second infinite family is based on an alternation of the regular family, given a Schläfli symbols h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of Coxeter group S_n (or C^~_n-1) for n>=4. A lower symmetry form Q_n (or B^~_n-1) can be created by removing another mirror on a order-4 peak.

The alternated hypercube facets become demihypercubes, and the deleted vertices create new cross-polytope facets. The vertex figure for honeycombs of this family are rectified hypercubes.

These are also named as - hδ_n for an (n-1)-dimensional honeycomb.

hδn	Name	Schläfli symbol	Coxeter-Dynkin diagrams Coxeter group
			Alternated regular h[4,3...,4]	Uniform-1 h[4,3...,31,1]	Uniform-2 P4 / Qn
hδ2	Apeirogon	{∞}
hδ3	Alternated square tiling (Same as regular square tiling {4,4})	h{4,4}
hδ4	Alternated cubic honeycomb	h{4,3,4}
hδ5	Alternated tesseractic tetracomb or demitesseractic tetracomb (Same as regular {3,3,4,3})	h{4,32,4}
hδ6	Demipenteractic pentacomb	h{4,33,4}
hδ7	Demihexeractic hexacomb	h{4,34,4}
hδ8	Demihepteractic heptacomb	h{4,35,4}
δ9	Demiocteractic octacomb	h{4,36,4}
δ10	Demienneractic enneacomb	h{4,37,4}	...

[edit] References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
  1. pp. 122-123, 1973. (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