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From Wikipedia, the free encyclopedia
In six-dimensional Euclidean geometry, the omnitruncated 6-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 6-simplex facets.
Omnitruncated 6-simplex honeycomb | |
---|---|
(No image) | |
Type | Uniform honeycomb |
Family | Omnitruncated simplectic honeycomb |
Schläfli symbol | {3[8]} |
Coxeter–Dynkin diagrams | |
Facets | t0,1,2,3,4,5{3,3,3,3,3} |
Vertex figure | Irr. 6-simplex |
Symmetry | ×14, [7[3[7]]] |
Properties | vertex-transitive |
The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
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[1] 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 |
Regular and uniform honeycombs in 6-space:
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