| 251 honeycomb | |
|---|---|
| (No image) | |
| Type | Uniform tessellation |
| Family | 2k1 polytope |
| Schläfli symbol | {3,3,35,1} |
| Coxeter symbol | 251 |
| Coxeter-Dynkin diagram |    
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| 8-face types | 241 {37} 
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| 7-face types | 231 {36} 
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| 6-face types | 221 {35} 
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| 5-face types | 211 {34} 
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| 4-face type | {33}
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| Cells | {32}
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| Faces | {3}
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| Edge figure | 051 
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| Vertex figure | 151 
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| Edge figure | 051 
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| Coxeter group |  , [35,2,1]
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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.
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.
| 2k1 figuresinn dimensions | |||||||||||
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| 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+
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E10 =  = E8++
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| Coxeter diagram |
   
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| 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 | 
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| Name | 2−1,1 | 201 | 211 | 221 | 231 | 241 | 251 | 261 | |||
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Fundamental convex regular and uniform honeycombs in dimensions 2–9
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| Space | Family | 
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| 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 | |
| E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
| En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |