8-simplex honeycomb

This vertex arrangement is called the A8 latticeor8-simplex lattice. The 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the ${\displaystyle {\tilde {A}}_{8}}$  Coxeter group.^[1] 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.

${\displaystyle {\tilde {E}}_{8}}$  contains ${\displaystyle {\tilde {A}}_{8}}$  as a subgroup of index 5760.^[2] Both ${\displaystyle {\tilde {E}}_{8}}$  and ${\displaystyle {\tilde {A}}_{8}}$  can be seen as affine extensions of ${\displaystyle A_{8}}$  from different nodes:  

The A³
₈ lattice is the union of three A₈ lattices, and also identical to the E8 lattice.^[3]

          ∪           ∪           =                .

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^[4] constructed by the ${\displaystyle {\tilde {A}}_{8}}$  Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:

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:

• Regular and uniform honeycombs in 8-space:
  • 8-cubic honeycomb
  • 8-demicubic honeycomb
  • Truncated 8-simplex honeycomb
  • 5₂₁ honeycomb
  • 2₅₁ honeycomb
  • 1₅₂ 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]