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 ${\displaystyle {\tilde {A}}_{7}}$  Coxeter group.^[1] 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.

${\displaystyle {\tilde {E}}_{7}}$  contains ${\displaystyle {\tilde {A}}_{7}}$  as a subgroup of index 144.^[2] Both ${\displaystyle {\tilde {E}}_{7}}$  and ${\displaystyle {\tilde {A}}_{7}}$  can be seen as affine extensions from ${\displaystyle A_{7}}$  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          .

This honeycomb is one of 29 unique uniform honeycombs^[3] constructed by the ${\displaystyle {\tilde {A}}_{7}}$  Coxeter group, grouped by their extended symmetry of rings within the regular octagon diagram:

Projection by folding edit

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:

• 7-cubic honeycomb
• 7-demicubic honeycomb
• Truncated 7-simplex honeycomb
• Omnitruncated 7-simplex honeycomb
• E7 honeycomb

• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley–Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]