2_{ 22} honeycomb
In geometry, the 2_{22} honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli 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.
2_{22} honeycomb | |
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(no image) | |
Type | Uniform tessellation |
Coxeter symbol | 2_{22} |
Schläfli symbol | {3,3,3^{2,2}} |
Coxeter diagram | |
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 | {3^{2,2}} |
Vertex figure | 1_{22} |
Coxeter group | , [[3,3,3<sup>2,2</sup>]] |
Properties | vertex-transitive, facet-transitive |
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,
Removing a node on the end of one of the 2-node branches leaves the 2_{21}, its only facet type,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 1_{22},
The edge figure is the vertex figure of the vertex figure, here being a birectified 5-simplex, t_{2}{3^{4}},
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 1_{22}.
E_{6} lattice
The 2_{22} honeycomb's vertex arrangement is called the E_{6} lattice.[1]
The E_{6}^{2} lattice, with [[3,3,3<sup>2,2</sup>]] symmetry, can be constructed by the union of two E_{6} lattices:
∪
The E_{6}^{*} lattice[2] (or E_{6}^{3}) with [3[3^{2,2,2}]] symmetry. The Voronoi cell of the E_{6}^{*} lattice is the rectified 1_{22} polytope, and the Voronoi tessellation is a bitruncated 2_{22} honeycomb.[3] It is constructed by 3 copies of the E_{6} lattice vertices, one from each of the three branches of the Coxeter diagram.
∪ ∪ = dual to .
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.
{3,3,3^{2,2}} | {3,3,4,3} |
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<sup>2,2</sup>]] with 2 equally ringed branches, 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: |
[[3,3,3<sup>2,2</sup>]] | ×2 |
24: |
[3[3^{2,2,2}]] | ×6 |
7: |
Birectified 2_{22} honeycomb
Birectified 2_{22} honeycomb | |
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(no image) | |
Type | Uniform tessellation |
Coxeter symbol | 0_{222} |
Schläfli symbol | {3^{2,2,2}} |
Coxeter diagram | |
6-face type | 0_{221} |
5-face types | 0_{22} 0_{211} |
4-face type | 0_{21} 24-cell 0_{111} |
Cell type | Tetrahedron 0_{20} Octahedron 0_{11} |
Face type | Triangle 0_{10} |
Vertex figure | Proprism {3}×{3}×{3} |
Coxeter group | 6×, [3[3^{2,2,2}]] |
Properties | vertex-transitive, facet-transitive |
The birectified 2_{22} honeycomb
Its facets are centered on the vertex arrangement of E_{6}^{*} lattice, as:
∪ ∪
Construction
The facet information can be extracted from its Coxeter–Dynkin diagram,
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes a proprism {3}×{3}×{3},
Removing a node on the end of one of the 3-node branches leaves the 1_{22}, its only facet type,
Removing a second end node defines 2 types of 5-faces: birectified 5-simplex, 0_{22} and birectified 5-orthoplex, 0_{211}.
Removing a third end node defines 2 types of 4-faces: rectified 5-cell, 0_{21}, and 24-cell, 0_{111}.
Removing a fourth end node defines 2 types of cells: octahedron, 0_{11}, and tetrahedron, 0_{20}.
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 paracompact 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 |
A_{2}A_{2} | E_{6} | =E_{6}^{+} | =E_{6}^{++} | |
Coxeter diagram |
|||||
Symmetry | [[3<sup>2,2,-1</sup>]] | [[3<sup>2,2,0</sup>]] | [[3<sup>2,2,1</sup>]] | [[3<sup>2,2,2</sup>]] | [[3<sup>2,2,3</sup>]] |
Order | 72 | 1440 | 103,680 | ∞ | |
Graph | ∞ | ∞ | |||
Name | −1_{22} | 0_{22} | 1_{22} | 2_{22} | 3_{22} |
The 2_{22} honeycomb is third in another dimensional series 2_{2k}.
Space | Finite | Euclidean | Hyperbolic | ||
---|---|---|---|---|---|
n | 4 | 5 | 6 | 7 | 8 |
Coxeter group |
A_{2}A_{2} | A_{5} | E_{6} | =E_{6}^{+} | E_{6}^{++} |
Coxeter diagram |
|||||
Graph | ∞ | ∞ | |||
Name | 2_{2,-1} | 2_{20} | 2_{21} | 2_{22} | 2_{23} |
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), Macmillan 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 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.
- Conway, John H.; Sloane, Neil J. A. (1998). Sphere Packings, Lattices and Groups ((3rd ed.) ed.). New York: Springer-Verlag. ISBN 0-387-98585-9. p125-126, 8.3 The 6-dimensional lattices: E6 and E6*
- Klitzing, Richard. "6D Hexacombs x3o3o3o3o *c3o3o - jakoh".
- Klitzing, Richard. "6D Hexacombs o3o3x3o3o *c3o3o - ramoh".
Fundamental convex regular and uniform honeycombs in dimensions 2-9 | ||||||
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Space | Family | / / | ||||
E^{2} | Uniform tiling | {3^{[3]}} | δ_{3} | hδ_{3} | qδ_{3} | Hexagonal |
E^{3} | Uniform convex honeycomb | {3^{[4]}} | δ_{4} | hδ_{4} | qδ_{4} | |
E^{4} | Uniform 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |
E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |
E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |
E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |
E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |
E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |
E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |