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 | {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 figure | t0,5{35} |
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 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
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 | ||
r14 | [7[3[7]]] | ×14 |
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 | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |