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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 | |
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(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.
The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.
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{3,3,32,2} | {3,3,4,3} |
The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with symmetry. 24 of them have doubled symmetry [[3,3,32,2]] 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 |
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[32,2,2] | Full |
8: |
[[3,3,32,2]] | ×2 |
24:
|
[[3,32,2,2]] | ×6 |
7: |
Birectified 222 honeycomb | |
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(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:
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 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.