6-simplex honeycomb(No image)TypeUniform 6-honeycombFamilySimplectic honeycombSchläfli symbol{3} = 0Coxeter diagram6-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 figuret0,5{35} Symmetry×2, Propertiesvertex-transitive In six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.
A6 lattice This vertex arrangement is called the A6 lattice or 6-simplex lattice. The 42 vertices of the expanded 6-simplex vertex figure represent the 42 roots of the Coxeter group. It is the 6-dimensional case of a simplectic honeycomb. Around each vertex figure are 126 facets: 7+7 6-simplex, 21+21 rectified 6-simplex, 35+35 birectified 6-simplex, with the count distribution from the 8th row of Pascal's triangle.
The A lattice (also called A) is the union of seven A6 lattices, and has the vertex arrangement of the dual to the omnitruncated 6-simplex honeycomb, and therefore the Voronoi cell of this lattice is the omnitruncated 6-simplex.
∪ ∪ ∪ ∪ ∪ ∪ = dual of
Related polytopes and honeycombs
Projection by folding
The 6-simplex honeycomb can be projected into the 3-dimensional cubic 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 6-space: 6-cubic honeycomb 6-demicubic honeycomb Truncated 6-simplex honeycomb Omnitruncated 6-simplex honeycomb 222 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]