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Concept
Hypercubic honeycomb
Summary
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.
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.
Official source
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