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6 cubic honeycomb

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The 6-cube honeycomb or hexeractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 6-space.

Contents

It is analogous to the square tiling of the plane and to the cubic honeycomb of 3-space.

Constructions

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,34,4}. Another form has two alternating 6-cube facets (like a checkerboard) with Schläfli symbol {4,33,31,1}. The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol {∞}6.

The [4,34,4], , Coxeter group generates 127 permutations of uniform tessellations, 71 with unique symmetry and 70 with unique geometry. The expanded 6-cubic honeycomb is geometrically identical to the 6-cubic honeycomb.

The 6-cubic honeycomb can be alternated into the 6-demicubic honeycomb, replacing the 6-cubes with 6-demicubes, and the alternated gaps are filled by 6-orthoplex facets.

Trirectified 6-cubic honeycomb

A trirectified 6-cubic honeycomb, , containins all birectified 6-orthoplex facets and is the Voronoi tessellation of the D6* lattice. Facets can be identically colored from a doubled C ~ 6 ×2, [[4,34,4]] symmetry, alternately colored from C ~ 6 , [4,34,4] symmetry, three colors from B ~ 6 , [4,33,31,1] symmetry, and 4 colors from D ~ 6 , [31,1,3,3,31,1] symmetry.

References

6-cubic honeycomb Wikipedia
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