| 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} |
| Vertex figure | 151 |
| Edge figure | 051 |
| Coxeter group | E ~ 8 {\displaystyle {\tilde {E}}_{8}} , [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.
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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
| 2k1 figures in n dimensions | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Space | Finite | Euclidean | Hyperbolic | ||||||||
| n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||
| Coxetergroup | E3=A2A1 | E4=A4 | E5=D5 | E6 | E7 | E8 | E9 = E ~ 8 {\displaystyle {\tilde {E}}_{8}} = E8+ | E10 = T ¯ 8 {\displaystyle {\bar {T}}_{8}} = E8++ | |||
| Coxeterdiagram | |||||||||||
| Symmetry | [3−1,2,1] | [30,2,1] | [[31,2,1]] | [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 | - | - | |||||||||
| Name | 2−1,1 | 201 | 211 | 221 | 231 | 241 | 251 | 261 | |||
- 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, wiley.com, ISBN 978-0-471-01003-6
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
| ||||||
|---|---|---|---|---|---|---|
| Space | Family | A ~ n − 1 {\displaystyle {\tilde {A}}_{n-1}} | C ~ n − 1 {\displaystyle {\tilde {C}}_{n-1}} | B ~ n − 1 {\displaystyle {\tilde {B}}_{n-1}} | D ~ n − 1 {\displaystyle {\tilde {D}}_{n-1}} | G ~ 2 {\displaystyle {\tilde {G}}_{2}} / F ~ 4 {\displaystyle {\tilde {F}}_{4}} / E ~ n − 1 {\displaystyle {\tilde {E}}_{n-1}} |
| 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 |