2_{51} honeycomb | |
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(No image) | |
Type | Uniform tessellation |
Family | 2_{k1} polytope |
Schläfli symbol | {3,3,3^{5,1}} |
Coxeter symbol | 2_{51} |
Coxeter-Dynkin diagram | |
8-face types | 2_{41} {3^{7}} |
7-face types | 2_{31} {3^{6}} |
6-face types | 2_{21} {3^{5}} |
5-face types | 2_{11} {3^{4}} |
4-face type | {3^{3}} |
Cells | {3^{2}} |
Faces | {3} |
Edge figure | 0_{51} |
Vertex figure | 1_{51} |
Edge figure | 0_{51} |
Coxeter group | , [3^{5,2,1}] |
In 8-dimensional geometry, the 2_{51} honeycomb is a space-filling uniform tessellation. It is composed of 2_{41} polytope and 8-simplex facets arranged in an 8-demicube vertex figure. It is the final figure in the 2_{k1} 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 2_{41}.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 8-demicube, 1_{51}.
The edge figure is the vertex figure of the vertex figure. This makes the rectified 7-simplex, 0_{51}.
2_{k1} figures in n dimensions | |||||||||||
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Space | Finite | Euclidean | Hyperbolic | ||||||||
n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||
Coxeter group |
E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||
Coxeter diagram |
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Symmetry | [3^{−1,2,1}] | [3^{0,2,1}] | [[3^{1,2,1}]] | [3^{2,2,1}] | [3^{3,2,1}] | [3^{4,2,1}] | [3^{5,2,1}] | [3^{6,2,1}] | |||
Order | 12 | 120 | 384 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ||||
Graph | - | - | |||||||||
Name | 2_{−1,1} | 2_{01} | 2_{11} | 2_{21} | 2_{31} | 2_{41} | 2_{51} | 2_{61} |
Fundamental convex regular and uniform honeycombs in dimensions 2-9
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Space | Family | / / | ||||
E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |