| bgcolor=#e7dcc3 colspan=2 | 222 honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | (no image) | |
| Type | Uniform tessellation | |
| Coxeter symbol | 222 | |
| Schläfli symbol | ||
| Coxeter diagram | ||
| 6-face type | 221 | |
| 5-face types | 211 | |
| 4-face type | ||
| Cell type | ||
| Face type | ||
| Face figure | × duoprism | |
| Edge figure | ||
| Vertex figure | 122 | |
| Coxeter group | {\tilde{E}}6 | |
| 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, .
The face figure is the vertex figure of the edge figure, here being a triangular duoprism, ×, .
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
{\tilde{E}}6
{\tilde{F}}4
{\tilde{E}}6 | {\tilde{F}}4 | |
|---|---|---|
The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with
{\tilde{E}}6
| Symmetry | Order | Honeycombs | |
|---|---|---|---|
| width=100 | [3<sup>2,2,2</sup>] | Full | 8:,,,,,,,. |
| 3,3,32,2 | ×2 | 24:,,,,,, ,,,,,, ,,,,,, ,,,,,. | |
| 3,32,2,2 | ×6 | 7:,,,,,,. |
| bgcolor=#e7dcc3 colspan=2 | Birectified 222 honeycomb | |
|---|---|---|
| bgcolor=#ffffff align=center colspan=2 | (no image) | |
| Type | Uniform tessellation | |
| Coxeter symbol | 0222 | |
| Schläfli symbol | ||
| 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 ×× | |
| Coxeter group | 6× {\tilde{E}}6 | |
| Properties | vertex-transitive, facet-transitive |
Its facets are centered on the vertex arrangement of E6* lattice, as:
∪ ∪
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 ××, .
Removing a node on the end of one of the 3-node branches leaves the rectified 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.
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.
The 222 honeycomb is third in another dimensional series 22k.
. John Horton Conway . Sloane, Neil J. A. . 1998 . Sphere Packings, Lattices and Groups . (3rd ed.) . Springer-Verlag . New York . 0-387-98585-9 . registration . Neil Sloane . p125-126, 8.3 The 6-dimensional lattices: E6 and E6*