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2 22 honeycomb
| 222 honeycomb |
|---|
| (no image) |
| Type |
| Coxeter symbol |
| Schläfli symbol |
| Coxeter diagram |
| 6-face type |
| 5-face types |
| 4-face type |
| Cell type |
| Face type |
| Face figure |
| Edge figure |
| Vertex figure |
| Coxeter group |
| Properties |
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, .
Removing a node on the end of one of the 2-node branches leaves the 221, its only facet type,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 122, .
The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t2{34}, .
The face figure is the vertex figure of the edge figure, here being a triangular duoprism, {3}×{3}, .
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.
The E62 lattice, with 3,3,32,2 symmetry, can be constructed by the union of two E6 lattices: : ∪
The E6 lattice* (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. It is constructed by 3 copies of the E6 lattice vertices, one from each of the three branches of the Coxeter diagram. : ∪ ∪ = dual to .
Geometric folding
The {\tilde{E}}_6 group is related to the {\tilde{F}}_4 by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.
| {\tilde{E}}_6 | {\tilde{F}}_4 | |
|---|---|---|
| {3,3,32,2} |
Notes
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, , 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.
- pp. 125–126, 8.3 The 6-dimensional lattices: E6 and E6*
References
- "The Lattice E6".
- "The Lattice E6*".
- [http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices] {{Webarchive. link. (2016-01-30 , Edward Pervin)
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