7 simplex 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 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.

E ~ 7 contains A ~ 7 as a subgroup of index 144. Both E ~ 7 and 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 .

Related polytopes and honeycombs

This honeycomb is one of 29 unique uniform honeycombs constructed by the 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: