6-simplex honeycomb
(No image)
TypeUniform 6-honeycomb
FamilySimplectic honeycomb
Schläfli symbol{3[7]} = 0[7]
Coxeter diagram
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 figuret0,5{35}
SymmetryA ~ 6 {\displaystyle {\tilde {A}}_{6}}×2, [[3[7]]]
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 A ~ 6 {\displaystyle {\tilde {A}}_{6}} 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* 6 lattice (also called A7 6) 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

This honeycomb is one of 17 unique uniform honeycombs constructed by the A ~ 6 {\displaystyle {\tilde {A}}_{6}} Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:

A6 honeycombs
Heptagon symmetryExtended symmetryExtended diagramExtended groupHoneycombs
a1[3[7]]A ~ 6 {\displaystyle {\tilde {A}}_{6}}
i2[[3[7]]]A ~ 6 {\displaystyle {\tilde {A}}_{6}}×21 2
r14[7[3[7]]]A ~ 6 {\displaystyle {\tilde {A}}_{6}}×143

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:

A ~ 6 {\displaystyle {\tilde {A}}_{6}}
C ~ 3 {\displaystyle {\tilde {C}}_{3}}

See also

Regular and uniform honeycombs in 6-space:

Notes

  • 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, ISBN978-0-471-01003-6 (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]
vteFundamental convex regular and uniform honeycombs in dimensions 2–9
SpaceFamilyA ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}}C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}}B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}}D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}}G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}}
E2Uniform tiling0[3]δ3hδ3qδ3Hexagonal
E3Uniform convex honeycomb0[4]δ4hδ4qδ4
E4Uniform 4-honeycomb0[5]δ5hδ5qδ524-cell honeycomb
E5Uniform 5-honeycomb0[6]δ6hδ6qδ6
E6Uniform 6-honeycomb0[7]δ7hδ7qδ7222
E7Uniform 7-honeycomb0[8]δ8hδ8qδ8133 • 331
E8Uniform 8-honeycomb0[9]δ9hδ9qδ9152 • 251 • 521
E9Uniform 9-honeycomb0[10]δ10hδ10qδ10
E10Uniform 10-honeycomb0[11]δ11hδ11qδ11
En−1Uniform (n−1)-honeycomb0[n]δnhδnqδn1k2 • 2k1 • k21