Hypercubic honeycomb
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 in -dimensional spaces with the Schläfli symbols and containing the symmetry of Coxeter group for.
The tessellation is constructed from 4 -hypercubes per ridge. The vertex figure is a cross-polytope
The hypercubic honeycombs are self-dual.
Coxeter named this family as for an -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.