1 33 honeycomb

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: [[File:Affine_A7_E7_relations.png]]

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 .

Alternative names

• Pentacontahexa-hecatonicosihexa-pentacosiheptacontahexa-exic heptacomb
• Rectified pentacontahexa-hecatonicosihexa-exic heptacomb
• Acronym: lanquoh (Jonathan Bowers)

Notes

References

• H.S.M. Coxeter, Regular Polytopes, 1973, 3rd edition, Dover, New York,
• Coxeter The Beauty of Geometry: Twelve Essays, Dover Publications, 1999, (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, wiley.com,
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• o3o3o3o3o3o3o *d3x - linoh, o3o3o3x3o3o3o *d3o - lanquoh

References

1. N.W. Johnson: ''Geometries and Transformations'', (2018) 12.4: Euclidean Coxeter groups, p.294
2. "The Lattice E7".
3. [http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices] {{Webarchive. link. (2016-01-30 , Edward Pervin)