# 6-cubic honeycomb

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

6-cubic honeycomb
(no image)
TypeRegular 6-honeycomb
Uniform 6-honeycomb
FamilyHypercube 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 figure8 {4,3,3}
(16-cell)
Vertex figure64 {4,34}
(6-orthoplex)
Coxeter group${\displaystyle {\tilde {C}}_{6}}$, [4,34,4]
${\displaystyle {\tilde {B}}_{6}}$, [4,33,31,1]
Dualself-dual
Propertiesvertex-transitive, edge-transitive, face-transitive, cell-transitive

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.

• List of regular polytopes

## 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
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
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 {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21