6-simplex honeycomb | |
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(No image) | |
Type | Uniform 6-honeycomb |
Family | Simplectic honeycomb |
Schläfli symbol | {3^{[7]}} = 0_{[7]} |
Coxeter diagram | |
6-face types | {3^{5}} , t_{1}{3^{5}} t_{2}{3^{5}} |
5-face types | {3^{4}} , t_{1}{3^{4}} t_{2}{3^{4}} |
4-face types | {3^{3}} , t_{1}{3^{3}} |
Cell types | {3,3} , t_{1}{3,3} |
Face types | {3} |
Vertex figure | t_{0,5}{3^{5}} |
Symmetry | ×2, [[3^{[7]}]] |
Properties | vertex-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.
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.^{[1]} 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 A^{7}
_{6}) is the union of seven A_{6} 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
This honeycomb is one of 17 unique uniform honeycombs^{[2]} constructed by the Coxeter group, grouped by their extended symmetry of the Coxeter–Dynkin diagrams:
A6 honeycombs | ||||
---|---|---|---|---|
Heptagon symmetry |
Extended symmetry |
Extended diagram |
Extended group |
Honeycombs |
a1 | [3^{[7]}] |
| ||
i2 | [[3^{[7]}]] | ×2 |
_{1} _{2} | |
r14 | [7[3^{[7]}]] | ×14 |
_{3} |
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:
Regular and uniform honeycombs in 6-space:
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E^{2} | Uniform tiling | 0_{[3]} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | 0_{[4]} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | 0_{[5]} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | 0_{[6]} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | 0_{[7]} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | 0_{[8]} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | 0_{[9]} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | 0_{[10]} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{10} | Uniform 10-honeycomb | 0_{[11]} | δ_{11} | hδ_{11} | qδ_{11} | |
E^{n-1} | Uniform (n-1)-honeycomb | 0_{[n]} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |