1 33 honeycomb

Summary

133 honeycomb
(no image)
Type Uniform tessellation
Schläfli symbol {3,33,3}
Coxeter symbol 133
Coxeter-Dynkin diagram
or
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 Schläfli symbol {3,33,3}, and is composed of 132 facets.

Construction

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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

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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

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The   group is related to the   by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.

   
                   
{3,33,3} {3,3,4,3}

E7* lattice

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  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:

                   =                                      = dual of          .
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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.

13k dimensional figures
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7  =E7+  =E7++
Coxeter
diagram
                                                                   
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [[33,3,1]] [34,3,1]
Order 48 720 23,040 2,903,040
Graph       - -
Name 13,-1 130 131 132 133 134

Rectified 133 honeycomb

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Rectified 133 honeycomb
(no image)
Type Uniform tessellation
Schläfli symbol {33,3,1}
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 {3,3}
{3,4}
Face type {3}
Vertex figure {}×{3,3}×{3,3}
Coxeter group  , [[3,33,3]]
Properties vertex-transitive, facet-transitive

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

See also

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Notes

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  1. ^ N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
  2. ^ "The Lattice E7".
  3. ^ The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, Edward Pervin

References

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  • 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]
  • Klitzing, Richard. "7D Heptacombs o3o3o3o3o3o3o *d3x - linoh".
  • Klitzing, Richard. "7D Heptacombs o3o3o3x3o3o3o *d3o - rolinoh".
Space Family           /   /  
E2 Uniform tiling 0[3] δ3 3 3 Hexagonal
E3 Uniform convex honeycomb 0[4] δ4 4 4
E4 Uniform 4-honeycomb 0[5] δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb 0[6] δ6 6 6
E6 Uniform 6-honeycomb 0[7] δ7 7 7 222
E7 Uniform 7-honeycomb 0[8] δ8 8 8 133331
E8 Uniform 8-honeycomb 0[9] δ9 9 9 152251521
E9 Uniform 9-honeycomb 0[10] δ10 10 10
E10 Uniform 10-honeycomb 0[11] δ11 11 11
En-1 Uniform (n-1)-honeycomb 0[n] δn n n 1k22k1k21