# 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)
TypeUniform tessellation
Coxeter symbol222
Schläfli symbol{3,3,32,2}
Coxeter diagram
6-face type221
5-face types211
{34}
4-face type{33}
Cell type{3,3}
Face type{3}
Face figure{3}×{3} duoprism
Edge figure{32,2}
Vertex figure122
Coxeter group${\displaystyle {\tilde {E}}_{6}}$, [[3,3,3<sup>2,2</sup>]]
Propertiesvertex-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.

## 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.[1]

The E62 lattice, with [[3,3,3<sup>2,2</sup>]] 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 .

## Geometric folding

The ${\displaystyle {\tilde {E}}_{6}}$ group is related to the ${\displaystyle {\tilde {F}}_{4}}$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.

${\displaystyle {\tilde {E}}_{6}}$${\displaystyle {\tilde {F}}_{4}}$
{3,3,32,2} {3,3,4,3}

The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with ${\displaystyle {\tilde {E}}_{6}}$ 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[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
[32,2,2] Full

8: , , , , , , , .

[[3,3,3<sup>2,2</sup>]] ×2

24: , , , , , ,

, , , , , ,

, , , , , ,

, , , , , .

[3[32,2,2]] ×6

7: , , , , , , .

### Birectified 222 honeycomb

Birectified 222 honeycomb
(no image)
TypeUniform tessellation
Coxeter symbol0222
Schläfli symbol{32,2,2}
Coxeter diagram
6-face type0221
5-face types022
0211
4-face type021
24-cell 0111
Cell typeTetrahedron 020
Octahedron 011
Face typeTriangle 010
Vertex figureProprism {3}×{3}×{3}
Coxeter group${\displaystyle {\tilde {E}}_{6}}$, [3[32,2,2]]
Propertiesvertex-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.

k22 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 E6 ${\displaystyle {\tilde {E}}_{6}}$=E6+ ${\displaystyle {\bar {T}}_{7}}$=E6++
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 122 022 122 222 322

The 222 honeycomb is third in another dimensional series 22k.

22k figures of n dimensions
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2 A5 E6 ${\displaystyle {\tilde {E}}_{6}}$=E6+ E6++
Coxeter
diagram
Graph
Name 22,-1 220 221 222 223

## 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) 345]
• 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
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$ ${\displaystyle {\tilde {C}}_{n-1}}$ ${\displaystyle {\tilde {B}}_{n-1}}$ ${\displaystyle {\tilde {D}}_{n-1}}$ ${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
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 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21