Английская Википедия:2 51 honeycomb

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Шаблон:Short description

251 honeycomb
(No image)
Type Uniform tessellation
Family 2k1 polytope
Schläfli symbol {3,3,35,1}
Coxeter symbol 251
Coxeter-Dynkin diagram Шаблон:CDD
8-face types 241Файл:Gosset 2 41 petrie.svg
{37}Файл:8-simplex t0.svg
7-face types 231Файл:Gosset 2 31 polytope.svg
{36}Файл:7-simplex t0.svg
6-face types 221Файл:E6 graph.svg
{35}Файл:6-simplex t0.svg
5-face types 211Файл:Cross graph 5.svg
{34}Файл:5-simplex t0.svg
4-face type {33}Файл:4-simplex t0.svg
Cells {32}Файл:3-simplex t0.svg
Faces {3}Файл:2-simplex t0.svg
Edge figure 051 Файл:6-simplex t1.svg
Vertex figure 151 Файл:8-demicube.svg
Edge figure 051 Файл:7-simplex t1.svg
Coxeter group <math>{\tilde{E}}_8</math>, [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.

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.

Шаблон:CDD

Removing the node on the short branch leaves the 8-simplex.

Шаблон:CDD

Removing the node on the end of the 5-length branch leaves the 241.

Шаблон:CDD

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 8-demicube, 151.

Шаблон:CDD

The edge figure is the vertex figure of the vertex figure. This makes the rectified 7-simplex, 051.

Шаблон:CDD

Шаблон:-

Related polytopes and honeycombs

Шаблон:2 k1 polytopes

References

  • Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, Шаблон:ISBN (Chapter 3: Wythoff's Construction for Uniform Polytopes)
  • Coxeter Regular Polytopes (1963), Macmillan Company
    • Regular Polytopes, Third edition, (1973), Dover edition, Шаблон:ISBN (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, Шаблон:ISBN [1]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]

Шаблон:Honeycombs