# 6-simplex honeycomb

6-simplex honeycomb
(No image)
Type Uniform 6-honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[7]}
Coxeter diagram
6-face types {35} , t1{35}
t2{35}
5-face types {34} , t1{34}
t2{34}
4-face types {33} , t1{33}
Cell types {3,3} , t1{3,3}
Face types {3}
Vertex figure t0,5{35}
Symmetry ${\displaystyle {\tilde {A}}_{6}}$×2, [[3[7]]]
Properties vertex-transitive

In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.

## A6 lattice

This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the ${\displaystyle {\tilde {A}}_{6}}$ Coxeter group.[1] It is the 6-dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6-simplex, 21+21 rectified 6-simplex, 35+35 birectified 6-simplex, with the count distribution from the 8th row of Pascal's triangle.

The A*
6
lattice (also called A7
6
) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.

= dual of

## Related polytopes and honeycombs

This honeycomb is one of 17 unique uniform honeycombs[2] constructed by the ${\displaystyle {\tilde {A}}_{6}}$ Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

## Projection by folding

The 6-simplex honeycomb can be projected into the 3-dimensional cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:

Regular and uniform honeycombs in 6-space:

## Notes

1. ^ http://www2.research.att.com/~njas/lattices/A6.html
2. ^ * , A000029 18-1 cases, skipping one with zero marks

## References

• Norman Johnson Uniform Polytopes, Manuscript (1991)
• 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 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
• (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 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21