3 31 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 the node on the short branch leaves the 6-simplex facet: :

Removing the node on the end of the 3-length branch leaves the 321 facet: :

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 231 polytope. :

The edge figure is determined by removing the ringed node and ringing the neighboring node. This makes 6-demicube (131). :

The face figure is determined by removing the ringed node and ringing the neighboring node. This makes rectified 5-simplex (031). :

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}. :

Kissing number

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 231.

E7 lattice

The 331 honeycomb's vertex arrangement is called the 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 can also be expressed as a union of the vertices of two A7 lattices, also called A72: : = ∪

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 honeycombs

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.

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, , GoogleBook
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• R. T. Worley, The Voronoi Region of E7*. SIAM J. Discrete Math., 1.1 (1988), 134–141.
• pp. 124–125, §8.2 The 7-dimensional lattices: E7 and E7*

References

1. "The Lattice E7".
2. N.W. Johnson: ''Geometries and Transformations'', (2018), Chapter 12: Euclidean symmetry groups, p. 177
3. "The Lattice E7".
4. [http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices] {{Webarchive. link. (2016-01-30 , Edward Pervin)