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 | , [[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.
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 group is related to the by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.
| Template:CDD | Template:CDD |
| {3,33,3} | {3,3,4,3} |
E7* lattice
contains as a subgroup of index 144.[1] Both and can be seen as affine extension from 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.
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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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- ↑ N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 12: Euclidean symmetry groups, p 177
- ↑ http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html
- ↑ The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin