2_{ 22} honeycomb
2_{22} honeycomb | |
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(no image) | |
Type | Uniform tessellation |
Coxeter symbol | 2_{22} |
Schläfli symbol | {3,3,3^{2,2}} |
Coxeter–Dynkin diagram | Template:CDD |
6-face type | 2_{21} |
5-face types | 2_{11} {3^{4}} |
4-face type | {3^{3}} |
Cell type | {3,3} |
Face type | {3} |
Face figure | {3}×{3} duoprism |
Edge figure | t_{2}{3^{4}} |
Vertex figure | 1_{22} |
Coxeter group|, [[3,3,3^{2,2}]] | |
Properties | vertex-transitive, facet-transitive |
In geometry, the 2_{22} honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schlafli symbol {3,3,3^{2,2}}. It is constructed from 2_{21} facets and has a 1_{22} vertex figure, with 54 2_{21} polytopes around every vertex.
Its vertex arrangement is the E_{6} lattice, and the root system of the E_{6} Lie group so it can also be called the E_{6} 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, Template:CDD.
Removing a node on the end of one of the 2-node branches leaves the 2_{21}, its only facet type, Template:CDD
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 1_{22}, Template:CDD.
The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t_{2}{3^{4}}, Template:CDD.
The face figure is the vertex figure of the edge figure, here being a triangular duoprism, {3}×{3}, Template: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 1_{22}.
E6 lattice
The 2_{22} honeycomb's vertex arrangement is called the E_{6} lattice.^{[1]}
The E_{6}^{2 lattice}, with [[3,3,3^{2,2}]] symmetry, can be constructed by the union of two E_{6} lattices:
The E_{6}^{*} lattice^{[2]} (or E_{6}^{3) 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. }
- Template:CDD + Template:CDD + Template:CDD = dual to Template:CDD.
Related honeycombs
The 2_{22} honeycomb is one of 127 uniform honeycombs (39 unique) with symmetry. 24 of them have doubled symmetry [[3,3,3^{2,2}]] with 2 equally ringed branches and, and 7 have sextupled (3!) symmetry [3[3^{2,2,2}]] with identical rings on all 3 branches. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but the 2_{22} and birectified 2_{22} are isotopic, with only one type of facet: 2_{21}, and rectified 1_{22} polytopes respectively.
Symmetry | Order | Honeycombs |
---|---|---|
[3^{2,2,2}] | Full |
8: Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD. |
[[3,3,3^{2,2}]] | ×2 |
24: Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD. |
[3[3^{2,2,2}]] | ×6 |
7: Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD. |
Bitruncated 2 22 honeycomb
The bitruncated 2_{22} honeycomb Template:CDD, has within its symmetry construction 3 copies of Template:CDD facets. Its vertex arrangement can also be constructed as an E_{6}^{*} lattice, as:
Geometric folding
The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.
Template:CDD | Template:CDD |
{3,3,3^{2,2}} | {3,3,4,3} |
k_{22} polytopes
The 2_{22} honeycomb, is fourth in a dimensional series of uniform polytopes, expressed by Coxeter as k_{22} series. The final is a noncompact hyperbolic honeycomb, 3_{22. Each progressive uniform polytope is constructed from the previous as its vertex figure. }
Space | Finite | Euclidean | Hyperbolic | ||
---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 |
Coxeter group |
2A_{2} | A_{5} | E_{6} | =E_{6}^{+} | =E_{6}^{++} |
Coxeter diagram |
Template:CDD | Template:CDD | Template:CDD | Template:CDD | Template:CDD |
Symmetry | [[3^{2,2,-1}]] | [[3^{2,2,0}]] | [[3^{2,2,1}]] | [[3^{2,2,2}]] | [[3^{2,2,3}]] |
Order | 72 | 1440 | 103,680 | ∞ | |
Graph | ∞ | ∞ | |||
Name | −1_{22} | 0_{22} | 1_{22} | 2_{22} | 3_{22} |
Notes
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), Macmillian 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 [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. (A), 43 (1987), 268-278.
- {{#invoke:citation/CS1|citation
|CitationClass=book }} p125-126, 8.3 The 6-dimensional lattices: E6 and E6*