Omnitruncated 8-simplex honeycomb
| 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 | [math]\displaystyle{ {\tilde{A}}_9 }[/math]×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).
A*8 lattice
The A*8 lattice (also called A98) 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[1] constructed by the [math]\displaystyle{ {\tilde{A}}_8 }[/math] 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]] |
|
[math]\displaystyle{ {\tilde{A}}_8 }[/math] |
|
| i2 | [[3[9]]] |     
|
[math]\displaystyle{ {\tilde{A}}_8 }[/math]×2 |
|
| i6 | [3[3[9]]] |   
|
[math]\displaystyle{ {\tilde{A}}_8 }[/math]×6 |
|
| r18 | [9[3[9]]] |
|
[math]\displaystyle{ {\tilde{A}}_8 }[/math]×18 | 3
|
See also
Regular and uniform honeycombs in 8-space:
- 8-cubic honeycomb
- 8-demicubic honeycomb
- 8-simplex honeycomb
- Truncated 8-simplex honeycomb
- 521 honeycomb
- 251 honeycomb
- 152 honeycomb
Notes
- ↑ * Weisstein, Eric W.. "Necklace". http://mathworld.wolfram.com/Necklace.html., OEIS sequence 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 [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]
Fundamental convex regular and uniform honeycombs in dimensions 2-9
| ||||||
|---|---|---|---|---|---|---|
| Space | Family | [math]\displaystyle{ {\tilde{A}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{C}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{B}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{D}}_{n-1} }[/math] | [math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math] |
| 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 | |
| En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |
 |  |
