# 7-cubic honeycomb

(Redirected from 7-cube honeycomb)
7-cubic honeycomb
(no image)
Type Regular 7-honeycomb
Uniform 7-honeycomb
Family Hypercube honeycomb
Schläfli symbol {4,35,4}
{4,34,31,1}
{∞}7
Coxeter-Dynkin diagrams

7-face type {4,3,3,3,3,3}
6-face type {4,3,3,3,3}
5-face type {4,3,3,3}
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 128 {4,35}
(7-orthoplex)
Coxeter group [4,35,4]
Dual self-dual
Properties vertex-transitive, edge-transitive, face-transitive, cell-transitive

The 7-cubic honeycomb or hepteractic honeycomb is the only regular space-filling tessellation (or honeycomb) in Euclidean 7-space.

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

There are many different Wythoff constructions of this honeycomb. The most symmetric form is regular, with Schläfli symbol {4,35,4}. Another form has two alternating 7-cube facets (like a checkerboard) with Schläfli symbol {4,34,31,1}. The lowest symmetry Wythoff construction has 128 types of facets around each vertex and a prismatic product Schläfli symbol {∞}7.

## Related honeycombs

The [4,35,4],                , Coxeter group generates 255 permutations of uniform tessellations, 135 with unique symmetry and 134 with unique geometry. The expanded 7-cubic honeycomb is geometrically identical to the 7-cubic honeycomb.

The 7-cubic honeycomb can be alternated into the 7-demicubic honeycomb, replacing the 7-cubes with 7-demicubes, and the alternated gaps are filled by 7-orthoplex facets.

A quadritruncated 7-cubic honeycomb,        , contains all tritruncated 7-orthoplex facets and is the Voronoi tessellation of the D7* lattice. Facets can be identically colored from a doubled ${\displaystyle {\tilde {C}}_{7}}$ ×2, [[4,35,4]] symmetry, alternately colored from ${\displaystyle {\tilde {C}}_{7}}$ , [4,35,4] symmetry, three colors from ${\displaystyle {\tilde {B}}_{7}}$ , [4,34,31,1] symmetry, and 4 colors from ${\displaystyle {\tilde {D}}_{7}}$ , [31,1,33,31,1] 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]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
${\displaystyle {\tilde {A}}_{n-1}}$  ${\displaystyle {\tilde {C}}_{n-1}}$  ${\displaystyle {\tilde {B}}_{n-1}}$  ${\displaystyle {\tilde {D}}_{n-1}}$  ${\displaystyle {\tilde {G}}_{2}}$  / ${\displaystyle {\tilde {F}}_{4}}$  / ${\displaystyle {\tilde {E}}_{n-1}}$
{3[3]} δ3 3 3 Hexagonal
{3[4]} δ4 4 4
{3[5]} δ5 5 5 24-cell honeycomb
{3[6]} δ6 6 6
{3[7]} δ7 7 7 222
{3[8]} δ8 8 8 133331
{3[9]} δ9 9 9 152251521
{3[10]} δ10 10 10
{3[n]} δn n n 1k22k1k21