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.

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 A$$ lattice can be constructed as the union of two A7 lattices, and is identical to the E7 lattice. ∪ =.

The A$$ lattice is the union of four A7 lattices, which is identical to the E7* lattice (or E$$). ∪ ∪  ∪  =  +  = dual of.

The A$$ lattice (also called A$$) 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.

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: