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.
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 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.
contains as a subgroup of index 144. Both and can be seen as affine extensions from from different nodes:
The A lattice can be constructed as the union of two A<sub>7</sub> lattices, and is identical to the E7 lattice.
⪠= .
The A lattice is the union of four A<sub>7</sub> lattices, which is identical to the E7* lattice (or E).
⪠⪠⪠= + = dual of .
The A lattice (also called A) is the union of eight A<sub>7</sub> 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 .
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:
Regular and uniform honeycombs in 7-space: