2_22_honeycomb
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.
...| 222 honeycomb | |
|---|---|
| (no image) | |
| Type | Uniform tessellation |
| Coxeter symbol | 222 |
| Schläfli symbol | {3,3,32,2} |
| Coxeter diagram |       |
| 6-face type | 221  |
| 5-face types | 211 {34}  |
| 4-face type | {33} |
| Cell type | {3,3} |
| Face type | {3} |
| Face figure | {3}×{3} duoprism |
| Edge figure | {32,2}  |
| Vertex figure | 122  |
| Coxeter group | , [[3,3,32,2]] |
| Properties | vertex-transitive, facet-transitive |
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.
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}, 
.
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.
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.
∪ ∪ = dual to .
The 
The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with
| Symmetry | Order | Honeycombs |
|---|---|---|
| [32,2,2] | Full |
8: |
| [[3,3,32,2]] | ×2 |
24:
|
| [[3,32,2,2]] | ×6 |
7: |
Birectified 222 honeycomb
...| Birectified 222 honeycomb | |
|---|---|
| (no image) | |
| Type | Uniform tessellation |
| Coxeter symbol | 0222 |
| Schläfli symbol | {32,2,2} |
| Coxeter diagram | |
| 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×, [[3,32,2,2]] |
| Properties | vertex-transitive, facet-transitive |
The birectified 222 honeycomb , has rectified 1 22 polytope facets, , 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, .
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 122, its only facet type, .
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.
, 
...| Space | Finite | Euclidean | Hyperbolic | ||
|---|---|---|---|---|---|
| n | 4 | 5 | 6 | 7 | 8 |
| Coxeter group |
A2A2 | E6 | =E6+ |
=E6++ | |
| Coxeter diagram |
|
 |
|
|
|
| Symmetry | [[32,2,-1]] | [[32,2,0]] | [[32,2,1]] | [[32,2,2]] | [[32,2,3]] |
| Order | 72 | 1440 | 103,680 | ∞ | |
| Graph | ∞ | ∞ | |||
| Name | −122 | 022 | 122 | 222 | 322 |
The 222 honeycomb is third in another dimensional series 22k.
- The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, Edward Pervin
- 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. Ser. 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 | ||||||
|---|---|---|---|---|---|---|
| Space | Family | |
|
 |
 |
 / /  |
| E2 | Uniform tiling | {3[3]} | δ3 | hδ3 | qδ3 | Hexagonal |
| E3 | Uniform convex honeycomb | {3[4]} | δ4 | hδ4 | qδ4 | |
| E4 | Uniform 4-honeycomb | {3[5]} | δ5 | hδ5 | qδ5 | 24-cell honeycomb |
| E5 | Uniform 5-honeycomb | {3[6]} | δ6 | hδ6 | qδ6 | |
| E6 | Uniform 6-honeycomb | {3[7]} | δ7 | hδ7 | qδ7 | 222 |
| E7 | Uniform 7-honeycomb | {3[8]} | δ8 | hδ8 | qδ8 | 133 • 331 |
| E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
| E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
| En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |