In 7-dimensional geometry, 1<sub>33</sub> is a uniform honeycomb, also given by Schläfli symbol {3,3<sup>3,3</sup>}, and is composed of 1<sub>32</sub> facets. It is also named pentacontahexa-hecatonicosihexa-exic heptacomb and Jonathan Bowers gives it acronym linoh
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 1<sub>32</sub>, 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, 0<sub>33</sub>.
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}ÃÂ{3,3}.
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.
The group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.
contains as a subgroup of index 144. Both and can be seen as affine extension from from different nodes:
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>:
The 1<sub>33</sub> is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1<sub>3k</sub> series. The final is a noncompact hyperbolic honeycomb, 1<sub>34</sub>.
The rectified 1<sub>33</sub> or 0<sub>331</sub>, Coxeter diagram has facets and , and vertex figure .