8-simplex honeycomb
| 8-simplex honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform honeycomb |
| Family | Simplectic honeycomb |
| Schläfli symbol | {3[9]} |
| Coxeter–Dynkin diagrams |      |
| 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 |
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 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.
- +
+ 
= 
.
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
This honeycomb is one of 45 unique uniform honeycombs[3] constructed by
the Coxeter group. The symmetry can be multiplied by the ring symmetry of the Coxeter diagrams:
| Enneagon symmetry |
Symmetry | Extended diagram |
Order | Honeycombs |
|---|---|---|---|---|
| a1 | [3[9]] | |
Full |
|
| i2 | [[3[9]]] |      |
×2 |
|
| d6 | [3[3[9]]] |    |
×6 | |
| r18 | [9[3[9]]] | |
×18 | 3 |
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:
|
|
|---|---|
 |
  |
See also
- Regular and uniform honeycombs in 8-space:
Notes
- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A8.html
- ↑ N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 12: Euclidean symmetry groups, p.177
- ↑
- Weisstein, Eric W., "Necklace", MathWorld., A000029 46-1 cases, skipping one with zero marks
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, ISBN 978-0-471-01003-6
- (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]
| Fundamental convex regular and uniform honeycombs in dimensions 2–11 | |||||
|---|---|---|---|---|---|
| Family |  |
 |
 |
 |
 /  /  |
| Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
| Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
| Uniform 5-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| Uniform 6-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
| Uniform 7-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
| Uniform 8-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
| Uniform 9-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| Uniform n-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |