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.
Contents
- Construction
- Kissing number
- E6 lattice
- Related honeycombs
- Birectified 2 22 honeycomb
- Geometric folding
- k22 polytopes
- References
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.
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 .Related honeycombs
The 222 honeycomb is one of 127 uniform honeycombs (39 unique) with
Birectified 2 22 honeycomb
The birectified 2 22 honeycomb , has within its symmetry construction 3 copies of facets. Its facets are centered on the vertex arrangement of E6* lattice, as:
∪ ∪Geometric folding
The
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.
The 222 honeycomb is third in another dimensional series 22k.