8-simplex honeycomb
8-simplex honeycomb |
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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.
A8 lattice
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 Ã8 Coxeter group. 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.
Ẽ8 contains Ã8 as a subgroup of index 5760. Both Ẽ8 and Ã8 can be seen as affine extensions of A8 from different nodes:
The A lattice is the union of three A8 lattices, and also identical to the E8 lattice.
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∪ ∪ = .
The A lattice (also called A) 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 .
Projection by folding
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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See also
- Regular and uniform honeycombs in 8-space:
- 8-cubic honeycomb
- 8-demicubic honeycomb
- Truncated 8-simplex honeycomb
- 521 honeycomb
- 251 honeycomb
- 152 honeycomb
References
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- Kaleidoscopes: Selected Writings of H. S. M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, 1
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]