6-cubic honeycomb

6-cubic honeycomb
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Type	Regular 6-honeycomb Uniform 6-honeycomb
Family	Hypercube honeycomb
Schläfli symbol	{4,34,4} {4,33,31,1}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
6-face type	{4,34}
5-face type	{4,33}
4-face type	{4,3,3}
Cell type	{4,3}
Face type	{4}
Face figure	{4,3} (octahedron)
Edge figure	8 {4,3,3} (16-cell)
Vertex figure	64 {4,34} (6-orthoplex)
Coxeter group	${\tilde{C}}_6$, [4,34,4] ${\tilde{B}}_6$, [4,33,31,1]
Dual	self-dual
Properties	vertex-transitive, edge-transitive, face-transitive, cell-transitive

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

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 {4,3³,3^1,1}. The lowest symmetry Wythoff construction has 64 types of facets around each vertex and a prismatic product Schläfli symbol {∞}⁽⁶⁾.

Related honeycombs

The [4,3⁴,4], Template:CDD, 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, Template:CDD, contains all birectified 6-orthoplex facets and is the Voronoi tessellation of the D₆^* lattice. Facets can be identically colored from a doubled ${\tilde{C}}_6$×2, [4,3⁴,4] symmetry, alternately colored from ${\tilde{C}}_6$, [4,3⁴,4] symmetry, three colors from ${\tilde{B}}_6$, [4,3³,3^1,1] symmetry, and 4 colors from ${\tilde{D}}_6$, [3^1,1,3,3,3^1,1] symmetry.

See also

• List of regular polytopes

References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover, p. 296, Table II: Regular honeycombs, Template:ISBN
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, Template:ISBN
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]