# 2 22 honeycomb

133 honeycomb
(no image)
Type Uniform tessellation
Schläfli symbol {3,33,3}
Coxeter symbol 133
Coxeter-Dynkin diagram Template:CDD
or Template:CDD
7-face type 132
6-face types 122
131
5-face types 121
{34}
4-face type 111
{33}
Cell type 101
Face type {3}
Cell figure Square
Face figure Triangular duoprism
Edge figure Tetrahedral duoprism
Vertex figure Trirectified 7-simplex
Coxeter group ${\displaystyle {\tilde {E}}_{7}}$, [[3,33,3]]
Properties vertex-transitive, facet-transitive

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schlafli symbol {3,33,3}, 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.

Template:CDD

Removing a node on the end of one of the 3-length branch leaves the 132, its only facet type.

Template:CDD

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the trirectified 7-simplex, 033.

Template:CDD

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}.

Template:CDD

## 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 ${\displaystyle {\tilde {E}}_{7}}$ 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.

## E7* lattice

${\displaystyle {\tilde {E}}_{7}}$ 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 [[3,33,3]]. 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:

Template:CDD + Template:CDD = Template:CDD + Template:CDD + Template:CDD + Template:CDD = dual of Template:CDD.

### 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. 3) On the lane approach area, make sure that you are able to slide nicely and that the shoe opposite the sliding shoe does not slip when you take your steps.
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## Notes

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## References

• 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]

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1. N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 12: Euclidean symmetry groups, p 177
2. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html
3. The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin