8 simplex honeycomb

A8 lattice

This vertex arrangement is called the A8 lattice or 8-simplex lattice. The 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the A ~ 8 Coxeter group. It is the 8-dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9 8-simplex, 36+36 rectified 8-simplex, 84+84 birectified 8-simplex, 126+126 trirectified 8-simplex, with the count distribution from the 10th row of Pascal's triangle.

E ~ 8 contains A ~ 8 as a subgroup of index 5760. Both E ~ 8 and A ~ 8 can be seen as affine extensions of A 8 from different nodes:

The A3
8 lattice is the union of three A₈ lattices, and also identical to the E8 lattice.

The A*
8 lattice (also called A9
8) is the union of nine A₈ lattices, and has the vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 8-simplex

∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ = dual of .

Related polytopes and honeycombs

This honeycomb is one of 45 unique uniform honeycombs constructed by the A ~ 8 Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:

Projection by folding

The 8-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: