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2 51 honeycomb

Eight-dimensional geometric tessellation

Eight-dimensional geometric tessellation

251 honeycomb
(No image)
Type
Family
Schläfli symbol
Coxeter symbol
Coxeter-Dynkin diagram
8-face types
7-face types
6-face types
5-face types
4-face type
Cells
Faces
Vertex figure
Edge figure
Coxeter group

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. :

References

  • Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, , (Chapter 3: Wythoff's Construction for Uniform Polytopes)
  • Coxeter Regular Polytopes (1963), Macmillan Company
    • Regular Polytopes, Third edition, (1973), Dover edition, , (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,
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
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9-polytopes e8-(mathematics) honeycombs-(geometry)
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