7-simplex honeycomb(No image)TypeUniform 7-honeycombFamilySimplectic honeycombSchläfli symbol{3} = 0Coxeter diagram6-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 figuret0,6{36} Symmetry×21, <[3]>Propertiesvertex-transitive 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 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.N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294 Both and can be seen as affine extensions from 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 .
Related polytopes and honeycombs
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:
See also Regular and uniform honeycombs in 7-space: 7-cubic honeycomb 7-demicubic honeycomb Truncated 7-simplex honeycomb Omnitruncated 7-simplex honeycomb E7 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]