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