# 2 22 honeycomb

Its vertex arrangement is the *E _{6} lattice*, and the root system of the E

_{6}Lie group so it can also be called the

**E**.

_{6}honeycombIt 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 2_{21}, its only facet type,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 1_{22}, .

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 1_{22}.

The **E _{6}^{2} lattice**, with [[3,3,3

^{2,2}]] symmetry, can be constructed by the union of two E

_{6}lattices:

The **E _{6}^{*} lattice**

^{[2]}(or E

_{6}

^{3}) with [3[3

^{2,2,2}]] symmetry. The Voronoi cell of the E

_{6}

^{*}lattice is the rectified 1

_{22}polytope, and the Voronoi tessellation is a bitruncated 2

_{22}honeycomb.

^{[3]}It is constructed by 3 copies of the E

_{6}lattice vertices, one from each of the three branches of the Coxeter diagram.

The facet information can be extracted from its Coxeter–Dynkin diagram, .

Removing a node on the end of one of the 3-node branches leaves the 1_{22}, its only facet type, .

Removing a second end node defines 2 types of 5-faces: birectified 5-simplex, 0_{22} and birectified 5-orthoplex, 0_{211}.

Removing a third end node defines 2 types of 4-faces: rectified 5-cell, 0_{21}, and 24-cell, 0_{111}.

Removing a fourth end node defines 2 types of cells: octahedron, 0_{11}, and tetrahedron, 0_{20}.

The 2_{22} honeycomb, is fourth in a dimensional series of uniform polytopes, expressed by Coxeter as k_{22} series. The final is a paracompact hyperbolic honeycomb, 3_{22}. Each progressive uniform polytope is constructed from the previous as its vertex figure.