Hypercubic honeycomb

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}.

The hypercubic honeycombs are self-dual.

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

Wythoff construction classes by dimension

There are 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 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 3 colors of cells around in vertex in 1:3:3:1 counts.

A more general class of honeycombs is 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.

See also

• Alternated hypercubic honeycomb

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