1 33 honeycomb

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets.

Construction
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 a node on the end of one of the 3-length branch leaves the 132, its only facet type.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 033.

The edge figure is determined by removing the ringed nodes of the vertex figure and ringing the neighboring node. This makes the tetrahedral duoprism, {3,3}×{3,3}.

Kissing number
Each vertex of this polytope corresponds to the center of a 6-sphere in a moderately dense sphere packing, in which each sphere is tangent to 70 others; the best known for 7 dimensions (the kissing number) is 126.

Geometric folding
The $${\tilde{E}}_7$$ group is related to the $${\tilde{F}}_4$$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.

E7* lattice
$${\tilde{E}}_7$$ contains $${\tilde{A}}_7$$ as a subgroup of index 144. Both $${\tilde{E}}_7$$ and $${\tilde{A}}_7$$ can be seen as affine extension from $$A_7$$ from different nodes:

The E7* lattice (also called E72) has double the symmetry, represented by 3,33,3. The Voronoi cell of the E7* lattice is the 132 polytope, and voronoi tessellation the 133 honeycomb. 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:
 * ∪ =  ∪  ∪  ∪  = dual of.

Related polytopes and honeycombs
The 133 is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The final is a noncompact hyperbolic honeycomb, 134.

Rectified 133 honeycomb
The rectified 133 or 0331, Coxeter diagram has facets  and, and vertex figure.