# 8-simplex honeycomb

In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, rectified 8-simplex, birectified 8-simplex, and trirectified 8-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb.

8-simplex honeycomb
(No image)
TypeUniform 8-honeycomb
FamilySimplectic honeycomb
Schläfli symbol{3}
Coxeter diagram
6-face types{37} , t1{37}
t2{37} , t3{37}
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 figuret0,7{37}
Symmetry${\tilde {A}}_{8}$ ×2, [[3]]
Propertiesvertex-transitive

## 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 ${\tilde {A}}_{8}$ 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.

${\tilde {E}}_{8}$ contains ${\tilde {A}}_{8}$ as a subgroup of index 5760. Both ${\tilde {E}}_{8}$ and ${\tilde {A}}_{8}$ 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 .

This honeycomb is one of 45 unique uniform honeycombs constructed by the ${\tilde {A}}_{8}$ 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: