E6 honeycomb

In geometry, an E6 honeycomb (or 222 honeycomb) is a tessellation of uniform polytopes in 6-dimensional Euclidean space.

{\tilde{E}}_6 is a affine Coxeter group. 127 uniform honeycombs can be generated from this family by all ring permutations of its Coxeter-Dynkin diagram. There are no regular honeycombs in the family since its Coxeter diagram a nonlinear graph, but there are one simplest ones, with a single ring at the end of one of its 3 branches: 222.

The 222 honeycomb's vertex arrangement is called the E6 lattice.[1]

Contents

222 honeycomb

222 honeycomb
(no image)
Type Uniform tessellation
Coxeter symbol 222
Schläfli symbol {3,3,32,2}
Coxeter–Dynkin 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 t2{34}
Vertex figure 122
Coxeter group {\tilde{E}}_6, [32,2,2]
Properties vertex-transitive, facet-transitive

The 222 honeycomb is a uniform tessellation. It can also be represented by the Schlafli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

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.

Geometric folding

The {\tilde{E}}_6 group is related to the {\tilde{F}}_4 by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.

{\tilde{E}}_6 {\tilde{F}}_4
{3,3,32,2} {3,3,4,3}

Notes

  1. ^ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/E6.html

References