# 7-simplex honeycomb

7-simplex honeycomb
(No image)
Type Uniform honeycomb
Family Simplectic honeycomb
Schläfli symbol {3[8]}
Coxeter diagram
6-face types {36} , t1{36}
t2{36} , t3{36}
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 t0,6{36}
Symmetry ${\tilde{A}}_7$×2, <[3[8]]>
Properties vertex-transitive

In seven-dimensional Euclidean geometry, the 7-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 7-simplex, rectified 7-simplex, birectified 7-simplex, and trirectified 7-simplex facets. These facet types occur in proportions of 2:2:2:1 respectively in the whole honeycomb.

## A7 lattice

This vertex arrangement is called the A7 lattice or 7-simplex lattice. The 56 vertices of the expanded 7-simplex vertex figure represent the 56 roots of the ${\tilde{A}}_7$ Coxeter group.[1] It is the 7-dimensional case of a simplectic honeycomb. Around each vertex figure are 254 facets: 8+8 7-simplex, 28+28 rectified 7-simplex, 56+56 birectified 7-simplex, 70 trirectified 7-simplex, with the count distribution from the 9th row of Pascal's triangle.

${\tilde{E}}_7$ contains ${\tilde{A}}_7$ as a subgroup of index 144.[2] Both ${\tilde{E}}_7$ and ${\tilde{A}}_7$ can be seen as affine extensions from $A_7$ from different nodes:

The A2
7
lattice can be constructed as the union of two A7 lattices, and is identical to the E7 lattice.

= .

The A4
7
lattice is the union of four A7 lattices, which is identical to the E7* lattice (or E2
7
).

.

The A*
7
lattice (also called A8
7
) is the union of eight A7 lattices, and has the vertex arrangement to the dual honeycomb of the omnitruncated 7-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 7-simplex.

= dual of .

## Related polytopes and honeycombs

This honeycomb is one of 29 unique uniform honeycombs[3] constructed by the ${\tilde{A}}_7$ Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:

Octagon
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycombs
a1 [3[8]] ×1

d2 <[3[8]]> ×2

1

p2 [[3[8]]] ×2
d4 <2[3[8]]> ×4

p4 [2[3[8]]] ×4

d8 [4[3[8]]] ×8
r16 [8[3[8]]] ×16 3

### Projection by folding

The 7-simplex honeycomb can be projected into the 4-dimensional tesseractic 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 7-space:

## Notes

1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A7.html
2. ^ N.W. Johnson: Geometries and Transformations, (2015) Chapter 12: Euclidean symmetry groups, p 177
3. ^ , A000029 30-1 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]