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In eighth-dimensional Euclidean geometry, the 8-simplex honeycomb is a space-filling tessellation (or honeycomb). The tessellation fills space by 8-simplex, rectified 8-simplex, birectified 8-simplex, and trirectified 8-simplex facets. These facet types occur in proportions of 1:1:1:1 respectively in the whole honeycomb.
8-simplex honeycomb | |
---|---|
(No image) | |
Type | Uniform 8-honeycomb |
Family | Simplectic honeycomb |
Schläfli symbol | {3[9]} = 0[9] |
Coxeter diagram | |
6-face types | {37} , t1{37} t2{37} , t3{37} |
6-face types | {36} , t1{36} t2{36} , t3{36} |
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 | t0,7{37} |
Symmetry | ×2, [[3[9]]] |
Properties | vertex-transitive |
This vertex arrangement is called the A8 lattice or 8-simplex lattice. The 72 vertices of the expanded 8-simplex vertex figure represent the 72 roots of the Coxeter group.[1] It is the 8-dimensional case of a simplectic honeycomb. Around each vertex figure are 510 facets: 9+9 8-simplex, 36+36 rectified 8-simplex, 84+84 birectified 8-simplex, 126+126 trirectified 8-simplex, with the count distribution from the 10th row of Pascal's triangle.
contains as a subgroup of index 5760.[2] Both and can be seen as affine extensions of from different nodes:
The A3
8 lattice is the union of three A8 lattices, and also identical to the E8 lattice.[3]
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[4] 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 | 3 |
The 8-simplex honeycomb can be projected into the 4-dimensional tesseractic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same vertex arrangement:
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