8-simplex honeycomb(No image)TypeUniform 8-honeycombFamilySimplectic honeycombSchläfli symbol{3} = 0Coxeter diagram6-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×2, Propertiesvertex-transitive 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.
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 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.
contains as a subgroup of index 5760.N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p.294 Both and can be seen as affine extensions of from different nodes:
The A lattice is the union of three A8 lattices, and also identical to the E8 lattice.Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950) ∪ ∪ = .
The A lattice (also called A) 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 .
Related polytopes and honeycombs
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:
See also Regular and uniform honeycombs in 8-space: 8-cubic honeycomb 8-demicubic honeycomb Truncated 8-simplex honeycomb 521 honeycomb 251 honeycomb 152 honeycomb
Notes
References 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, (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]