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.

The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes.

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 231.

The 331 honeycomb's vertex arrangement is called the E7 lattice.

[4] The E7* lattice is constructed by 2 copies of the E7 lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A7* lattices, also called A74: It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series.

A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.