8-simplex honeycomb
8-simplex honeycomb | |
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(No image) | |
Type | Uniform 8-honeycomb |
Family | Simplectic honeycomb |
Schläfli symbol | {3[9]} = 0[9] |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-face types | {37} ![]() ![]() t2{37} ![]() ![]() |
6-face types | {36} ![]() ![]() t2{36} ![]() ![]() |
6-face types | {35} ![]() ![]() t2{35} ![]() |
5-face types | {34} ![]() ![]() t2{34} ![]() |
4-face types | {33} ![]() ![]() |
Cell types | {3,3} ![]() ![]() |
Face types | {3} ![]() |
Vertex figure | t0,7{37} ![]() |
Symmetry | ×2, [[3[9]]] |
Properties | vertex-transitive |
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
[edit]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
.
Related polytopes and honeycombs
[edit]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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Projection by folding
[edit]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:
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
See also
[edit]- Regular and uniform honeycombs in 8-space:
Notes
[edit]- ^ "The Lattice A8".
- ^ N.W. Johnson: Geometries and Transformations, (2018) Chapter 12: Euclidean symmetry groups, p.294
- ^ Kaleidoscopes: Selected Writings of H. S. M. Coxeter, Paper 18, "Extreme forms" (1950)
- ^ * Weisstein, Eric W. "Necklace". MathWorld., OEIS sequence A000029 46-1 cases, skipping one with zero marks
References
[edit]- 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, ISBN 978-0-471-01003-6 [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]
Space | Family | / / | ||||
---|---|---|---|---|---|---|
E2 | Uniform tiling | 0[3] | δ3 | hδ3 | qδ3 | Hexagonal |
E3 | Uniform convex honeycomb | 0[4] | δ4 | hδ4 | qδ4 | |
E4 | Uniform 4-honeycomb | 0[5] | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
E5 | Uniform 5-honeycomb | 0[6] | δ6 | hδ6 | qδ6 | |
E6 | Uniform 6-honeycomb | 0[7] | δ7 | hδ7 | qδ7 | 222 |
E7 | Uniform 7-honeycomb | 0[8] | δ8 | hδ8 | qδ8 | 133 • 331 |
E8 | Uniform 8-honeycomb | 0[9] | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
E9 | Uniform 9-honeycomb | 0[10] | δ10 | hδ10 | qδ10 | |
E10 | Uniform 10-honeycomb | 0[11] | δ11 | hδ11 | qδ11 | |
En-1 | Uniform (n-1)-honeycomb | 0[n] | δn | hδn | qδn | 1k2 • 2k1 • k21 |