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2 51 honeycomb

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Eight-dimensional geometric tessellation
251 honeycomb
(No image)
Type Uniform tessellation
Family 2k1 polytope
Schläfli symbol {3,3,3}
Coxeter symbol 251
Coxeter-Dynkin diagram
8-face types 241
{3}
7-face types 231
{3}
6-face types 221
{3}
5-face types 211
{3}
4-face type {3}
Cells {3}
Faces {3}
Edge figure 051
Vertex figure 151
Edge figure 051
Coxeter group E ~ 8 {\displaystyle {\tilde {E}}_{8}} ,

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

2k1 figures in n dimensions
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 = E ~ 8 {\displaystyle {\tilde {E}}_{8}} = E8 E10 = T ¯ 8 {\displaystyle {\bar {T}}_{8}} = E8
Coxeter
diagram
Symmetry ]
Order 12 120 384 51,840 2,903,040 696,729,600 ∞
Graph - -
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,
Fundamental convex regular and uniform honeycombs in dimensions 2–9
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}}
E Uniform tiling 0 δ3 hδ3 qδ3 Hexagonal
E Uniform convex honeycomb 0 δ4 hδ4 qδ4
E Uniform 4-honeycomb 0 δ5 hδ5 qδ5 24-cell honeycomb
E Uniform 5-honeycomb 0 δ6 hδ6 qδ6
E Uniform 6-honeycomb 0 δ7 hδ7 qδ7 222
E Uniform 7-honeycomb 0 δ8 hδ8 qδ8 133 • 331
E Uniform 8-honeycomb 0 δ9 hδ9 qδ9 152 • 251 • 521
E Uniform 9-honeycomb 0 δ10 hδ10 qδ10
E Uniform 10-honeycomb 0 δ11 hδ11 qδ11
E Uniform (n-1)-honeycomb 0 δn hδn qδn 1k2 • 2k1 • k21
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