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}.

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 ${\displaystyle {\tilde {E}}_{7}}$  group is related to the ${\displaystyle {\tilde {F}}_{4}}$  by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.

${\displaystyle {\tilde {E}}_{7}}$  contains ${\displaystyle {\tilde {A}}_{7}}$  as a subgroup of index 144.^[1] Both ${\displaystyle {\tilde {E}}_{7}}$  and ${\displaystyle {\tilde {A}}_{7}}$  can be seen as affine extension from ${\displaystyle A_{7}}$  from different nodes:  

The E₇^* lattice (also called E₇²)^[2] has double the symmetry, represented by [[3,3^3,3]]. The Voronoi cell of the E₇^* lattice is the 1₃₂ polytope, and voronoi tessellation the 1₃₃ honeycomb.^[3] 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 edit

The 1₃₃ is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_3k series. The final is a noncompact hyperbolic honeycomb, 1₃₄.

Rectified 1₃₃ honeycomb edit

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

• 8-polytope
• 3₃₁ honeycomb

• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, ISBN 978-0-486-40919-1 (Chapter 3: Wythoff's Construction for Uniform Polytopes)
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• Klitzing, Richard. "7D Heptacombs o3o3o3o3o3o3o *d3x - linoh".
• Klitzing, Richard. "7D Heptacombs o3o3o3x3o3o3o *d3o - rolinoh".