1 33 honeycomb explained

bgcolor=#e7dcc3 colspan=2	133 honeycomb
bgcolor=#ffffff align=center colspan=2	(no image)
Type	Uniform tessellation
Schläfli symbol
Coxeter symbol	133
Coxeter-Dynkin diagram	or
7-face type	132
6-face types	122 131
5-face types	121
4-face type	111
Cell type	101
Face type
Cell figure	Square
Face figure	Triangular duoprism
Edge figure	Tetrahedral duoprism
Vertex figure
Coxeter group	{\tilde{E}} _{7} , 3,3^3,3
Properties	vertex-transitive, facet-transitive

In 7-dimensional geometry, 1₃₃ is a uniform honeycomb, also given by Schläfli symbol, 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, ×.

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.

{\tilde{E}} _{7}	{\tilde{F}} _{4}

E₇^* lattice

{\tilde{E}} _{7} contains {\tilde{A}} _{7} as a subgroup of index 144.^[1] 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)^[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

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

bgcolor=#e7dcc3 colspan=2	Rectified 133 honeycomb
bgcolor=#ffffff align=center colspan=2	(no image)
Type	Uniform tessellation
Schläfli symbol
Coxeter symbol	0331
Coxeter-Dynkin diagram	or
7-face type	Trirectified 7-simplex Rectified 1_32
6-face types	Birectified 6-simplex Birectified 6-cube Rectified 1_22
5-face types	Rectified 5-simplex Birectified 5-simplex Birectified 5-orthoplex
4-face type	5-cell Rectified 5-cell 24-cell
Cell type
Face type
Vertex figure	××
Coxeter group	{\tilde{E}} _{7} , 3,3^3,3
Properties	vertex-transitive, facet-transitive

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

See also

• 8-polytope
• 3₃₁ honeycomb

References

• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]

Notes and References

1. N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
2. Web site: The Lattice E7.
3. http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices