Английская Википедия:2 22 honeycomb
222 honeycomb | |
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(no image) | |
Type | Uniform tessellation |
Coxeter symbol | 222 |
Schläfli symbol | {3,3,32,2} |
Coxeter diagram | Шаблон:CDD |
6-face type | 221 Файл:E6 graph.svg |
5-face types | 211Файл:5-orthoplex.svg {34}Файл:5-simplex t0.svg |
4-face type | {33}Файл:4-simplex t0.svg |
Cell type | {3,3}Файл:3-simplex t0.svg |
Face type | {3}Файл:2-simplex t0.svg |
Face figure | {3}×{3} duoprism |
Edge figure | {32,2} Файл:5-simplex t2.svg |
Vertex figure | 122 Файл:Gosset 1 22 polytope.svg |
Coxeter group | <math>{\tilde{E}}_6</math>, [[3,3,32,2]] |
Properties | vertex-transitive, facet-transitive |
In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.
Its vertex arrangement is the E6 lattice, and the root system of the E6 Lie group so it can also be called the E6 honeycomb.
Construction
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 6-dimensional space.
The facet information can be extracted from its Coxeter–Dynkin diagram, Шаблон:CDD.
Removing a node on the end of one of the 2-node branches leaves the 221, its only facet type, Шаблон:CDD
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 122, Шаблон:CDD.
The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t2{34}, Шаблон:CDD.
The face figure is the vertex figure of the edge figure, here being a triangular duoprism, {3}×{3}, Шаблон:CDD.
Kissing number
Each vertex of this tessellation is the center of a 5-sphere in the densest known packing in 6 dimensions, with kissing number 72, represented by the vertices of its vertex figure 122.
E6 lattice
The 222 honeycomb's vertex arrangement is called the E6 lattice.[1]
The E62 lattice, with [[3,3,32,2]] symmetry, can be constructed by the union of two E6 lattices:
The E6* lattice[2] (or E63) with [[3,32,2,2]] symmetry. The Voronoi cell of the E6* lattice is the rectified 122 polytope, and the Voronoi tessellation is a bitruncated 222 honeycomb.[3] It is constructed by 3 copies of the E6 lattice vertices, one from each of the three branches of the Coxeter diagram.
- Шаблон:CDD ∪ Шаблон:CDD ∪ Шаблон:CDD = dual to Шаблон:CDD.
Geometric folding
The <math>{\tilde{E}}_6</math> group is related to the <math>{\tilde{F}}_4</math> by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.
<math>{\tilde{E}}_6</math> | <math>{\tilde{F}}_4</math> |
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Шаблон:CDD | Шаблон:CDD |
{3,3,32,2} | {3,3,4,3} |
Related honeycombs
The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with <math>{\tilde{E}}_6</math> symmetry. 24 of them have doubled symmetry [[3,3,32,2]] with 2 equally ringed branches, and 7 have sextupled (3!) symmetry [[3,32,2,2]] with identical rings on all 3 branches. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but the 222 and birectified 222 are isotopic, with only one type of facet: 221, and rectified 122 polytopes respectively.
Symmetry | Order | Honeycombs |
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[32,2,2] | Full |
8: Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD. |
[[3,3,32,2]] | ×2 |
24: Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD. |
[[3,32,2,2]] | ×6 |
7: Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD, Шаблон:CDD. |
Birectified 222 honeycomb
Birectified 222 honeycomb | |
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(no image) | |
Type | Uniform tessellation |
Coxeter symbol | 0222 |
Schläfli symbol | {32,2,2} |
Coxeter diagram | Шаблон:CDD |
6-face type | 0221 |
5-face types | 022 0211 |
4-face type | 021 24-cell 0111 |
Cell type | Tetrahedron 020 Octahedron 011 |
Face type | Triangle 010 |
Vertex figure | Proprism {3}×{3}×{3} |
Coxeter group | 6×<math>{\tilde{E}}_6</math>, [[3,32,2,2]] |
Properties | vertex-transitive, facet-transitive |
The birectified 222 honeycomb Шаблон:CDD, has rectified 1 22 polytope facets, Шаблон:CDD, and a proprism {3}×{3}×{3} vertex figure.
Its facets are centered on the vertex arrangement of E6* lattice, as:
Construction
The facet information can be extracted from its Coxeter–Dynkin diagram, Шаблон:CDD.
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes a proprism {3}×{3}×{3}, Шаблон:CDD.
Removing a node on the end of one of the 3-node branches leaves the 122, its only facet type, Шаблон:CDD.
Removing a second end node defines 2 types of 5-faces: birectified 5-simplex, 022 and birectified 5-orthoplex, 0211.
Removing a third end node defines 2 types of 4-faces: rectified 5-cell, 021, and 24-cell, 0111.
Removing a fourth end node defines 2 types of cells: octahedron, 011, and tetrahedron, 020.
k22 polytopes
The 222 honeycomb, is fourth in a dimensional series of uniform polytopes, expressed by Coxeter as k22 series. The final is a paracompact hyperbolic honeycomb, 322. Each progressive uniform polytope is constructed from the previous as its vertex figure. Шаблон:K 22 polytopes
The 222 honeycomb is third in another dimensional series 22k. Шаблон:2 2k polytopes
Notes
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] GoogleBook
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
- R. T. Worley, The Voronoi Region of E6*. J. Austral. Math. Soc. Ser. A, 43 (1987), 268–278.
- Шаблон:Cite book p125-126, 8.3 The 6-dimensional lattices: E6 and E6*
- Шаблон:KlitzingPolytopes
- Шаблон:KlitzingPolytopes