# 3 31 honeycomb

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 7-dimensional space.

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

Removing the node on the end of the 3-length branch leaves the **3 _{21}** facet:

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes **2 _{31}** polytope.

The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (**1 _{31}**).

The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (**0 _{31}**).

Each vertex of this tessellation is the center of a 6-sphere in the densest known packing in 7 dimensions; its kissing number is 126, represented by the vertices of its vertex figure 2_{31}.

The E_{7} lattice can also be expressed as a union of the vertices of two A_{7} lattices, also called A_{7}^{2}:

The **E _{7}^{*} lattice** (also called E

_{7}

^{2})

^{[3]}has double the symmetry, represented by [[3,3

^{3,3}]]. The Voronoi cell of the E

_{7}

^{*}lattice is the 1

_{32}polytope, and voronoi tessellation the 1

_{33}honeycomb.

^{[4]}The

**E**is constructed by 2 copies of the E

_{7}^{*}lattice_{7}lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A

_{7}

^{*}lattices, also called A

_{7}

^{4}:

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_{k1} series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.