# 7-simplex honeycomb

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.

7-simplex honeycomb
(No image)
TypeUniform 7-honeycomb
FamilySimplectic 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 figuret0,6{36}
Symmetry${\displaystyle {\tilde {A}}_{7}}$×21, <[3[8]]>
Propertiesvertex-transitive

## 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 ${\displaystyle {\tilde {A}}_{7}}$ 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.

${\displaystyle {\tilde {E}}_{7}}$ contains ${\displaystyle {\tilde {A}}_{7}}$ as a subgroup of index 144.[2] Both ${\displaystyle {\tilde {E}}_{7}}$ and ${\displaystyle {\tilde {A}}_{7}}$ can be seen as affine extensions from ${\displaystyle A_{7}}$ 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
).

= + = dual of .

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 .

This honeycomb is one of 29 unique uniform honeycombs[3] constructed by the ${\displaystyle {\tilde {A}}_{7}}$ Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:

### 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, (2018) 12.4: Euclidean Coxeter groups, p.294
3. Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence 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
• (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 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