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Concept
2 22 honeycomb
Summary
DISPLAYTITLE: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.
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.
The E62 lattice, with [[3,3,32,2]] symmetry, can be constructed by the union of two E6 lattices:
∪
The E6* lattice (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. 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 group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional 16-cell honeycomb.
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.
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