# 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}
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${\tilde {A}}_{7}$ ×21, <[3]>
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 ${\tilde {A}}_{7}$ Coxeter group. 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. 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
).

= + = 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 constructed by the ${\tilde {A}}_{7}$ 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: