In 7-dimensional geometry, the 3<sub>31</sub> honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3<sup>3,1</sup>} and is composed of 3<sub>21</sub> and 7-simplex facets, with 56 and 576 of them respectively around each vertex.
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 short branch leaves the 6-simplex facet:
Removing the node on the end of the 3-length branch leaves the 3<sub>21</sub> facet:
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 2<sub>31</sub> polytope.
The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (1<sub>31</sub>).
The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (0<sub>31</sub>).
The cell figure is determined by removing the ringed node of the face figure and ringing the neighboring nodes. This makes tetrahedral prism {}×{3,3}.
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<sub>31</sub>.
The 3<sub>31</sub> honeycomb's vertex arrangement is called the E<sub>7</sub> lattice.
contains as a subgroup of index 144. Both and can be seen as affine extension from from different nodes:
The E<sub>7</sub> lattice can also be expressed as a union of the vertices of two A<sub>7</sub> lattices, also called A<sub>7</sub><sup>2</sup>:
The E<sub>7</sub><sup>*</sup> lattice (also called E<sub>7</sub><sup>2</sup>) has double the symmetry, represented by 3,3<sup>3,3</sup>. The Voronoi cell of the E<sub>7</sub><sup>*</sup> lattice is the 1<sub>32</sub> polytope, and voronoi tessellation the 1<sub>33</sub> honeycomb. The E<sub>7</sub><sup>*</sup> lattice is constructed by 2 copies of the E<sub>7</sub> lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A<sub>7</sub><sup>*</sup> lattices, also called A<sub>7</sub><sup>4</sup>:
It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3<sub>k1</sub> series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.