Omnitruncated 8-simplex honeycomb | |
---|---|
(No image) | |
Type | Uniform honeycomb |
Family | Omnitruncated simplectic honeycomb |
Schläfli symbol | {3[9]} |
Coxeter–Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-face types | t01234567{3,3,3,3,3,3,3} |
Vertex figure |
![]() Irr. 8-simplex |
Symmetry | ×18, [9[3[9]]] |
Properties | vertex-transitive |
In eight-dimensional Euclidean geometry, the omnitruncated 8-simplex honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 8-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*
8 lattice (also called A9
8) is the union of nine A8 lattices, and has the
vertex arrangement of the dual honeycomb to the omnitruncated 8-simplex honeycomb, and therefore the
Voronoi cell of this lattice is an
omnitruncated 8-simplex
∪
∪
∪
∪
∪
∪
∪
∪
= dual of
.
This honeycomb is one of 45 unique uniform honeycombs [1] constructed by the Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:
A8 honeycombs | ||||
---|---|---|---|---|
Enneagon symmetry |
Symmetry | Extended diagram |
Extended group |
Honeycombs |
a1 | [3[9]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
| |
i2 | [[3[9]]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×2 |
|
i6 | [3[3[9]]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×6 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
r18 | [9[3[9]]] | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
×18 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Regular and uniform honeycombs in 8-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 |