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6-cubic honeycomb

## Summary

6-cubic honeycomb
(no image)
Type Regular 6-honeycomb
Uniform 6-honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,34,4}
{4,33,31,1}
Coxeter-Dynkin diagrams

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 ${\displaystyle {\tilde {C}}_{6}}$, [4,34,4]
${\displaystyle {\tilde {B}}_{6}}$, [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 {4,33,31,1}. 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,        , contains all birectified 6-orthoplex facets and is the Voronoi tessellation of the D6* lattice. Facets can be identically colored from a doubled ${\displaystyle {\tilde {C}}_{6}}$ ×2, [[4,34,4]] symmetry, alternately colored from ${\displaystyle {\tilde {C}}_{6}}$ , [4,34,4] symmetry, three colors from ${\displaystyle {\tilde {B}}_{6}}$ , [4,33,31,1] symmetry, and 4 colors from ${\displaystyle {\tilde {D}}_{6}}$ , [31,1,3,3,31,1] symmetry.

## References

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$  ${\displaystyle {\tilde {C}}_{n-1}}$  ${\displaystyle {\tilde {B}}_{n-1}}$  ${\displaystyle {\tilde {D}}_{n-1}}$  ${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling 0[3] δ3 3 3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 4 4
E4 Uniform 4-honeycomb 0[5] δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 6 6
E6 Uniform 6-honeycomb 0[7] δ7 7 7 222
E7 Uniform 7-honeycomb 0[8] δ8 8 8 133331
E8 Uniform 8-honeycomb 0[9] δ9 9 9 152251521
E9 Uniform 9-honeycomb 0[10] δ10 10 10
E10 Uniform 10-honeycomb 0[11] δ11 11 11
En-1 Uniform (n-1)-honeycomb 0[n] δn n n 1k22k1k21