2 51 honeycomb
| 251 honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform tessellation |
| Family | 2k1 polytope |
| Schläfli symbol | {3,3,35,1} |
| Coxeter symbol | 251 |
| Coxeter-Dynkin diagram |     |
| 8-face types | 241 {37}  |
| 7-face types | 231 {36}  |
| 6-face types | 221 {35}  |
| 5-face types | 211 {34}  |
| 4-face type | {33} |
| Cells | {32} |
| Faces | {3} |
| Edge figure | 051  |
| Vertex figure | 151  |
| Edge figure | 051  |
| Coxeter group |  , [35,2,1] |
In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation. It is composed of 241 polytope and 8-simplex facets arranged in an 8-demicube vertex figure. It is the final figure in the 2k1 family.
Construction
It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram.
Removing the node on the short branch leaves the 8-simplex.
Removing the node on the end of the 5-length branch leaves the 241.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 8-demicube, 151.
The edge figure is the vertex figure of the vertex figure. This makes the rectified 7-simplex, 051.
Related polytopes and honeycombs
| Space | Finite | Euclidean | Hyperbolic | |||||
|---|---|---|---|---|---|---|---|---|
| n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Coxeter group |
E3=A2A1 | E4=A4 | E5=D5 | E6 | E7 | E8 | E9 =  = E8+ |
E10 =  = E8++ |
| Coxeter diagram |
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| Symmetry | [3−1,2,1] | [30,2,1] | [[3<sup>1,2,1</sup>]] | [32,2,1] | [33,2,1] | [34,2,1] | [35,2,1] | [36,2,1] |
| Order | 12 | 120 | 384 | 51,840 | 2,903,040 | 696,729,600 | ∞ | |
| Graph |  |
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| Name | 2-1,1 | 201 | 211 | 221 | 231 | 241 | 251 | 261 |
References
- Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
- Coxeter Regular Polytopes (1963), Macmillan Company
- Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (Chapter 5: The Kaleidoscope)
- 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 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 2–10 | |||||
|---|---|---|---|---|---|
| Family |  |
 |
 |
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 /  /  |
| 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 |