# 1 33 honeycomb

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets.

133 honeycomb
(no image)
TypeUniform tessellation
Schläfli symbol{3,33,3}
Coxeter symbol133
Coxeter-Dynkin diagram
or
7-face type132
6-face types122
131
5-face types121
{34}
4-face type111
{33}
Cell type101
Face type{3}
Cell figureSquare
Face figureTriangular duoprism
Edge figureTetrahedral duoprism
Vertex figureTrirectified 7-simplex
Coxeter group${\displaystyle {\tilde {E}}_{7}}$, [[3,3<sup>3,3</sup>]]
Propertiesvertex-transitive, facet-transitive

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

${\displaystyle {\tilde {E}}_{7}}$${\displaystyle {\tilde {F}}_{4}}$
{3,33,3} {3,3,4,3}

## 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 [[3,3<sup>3,3</sup>]]. 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 .

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 ${\displaystyle {\tilde {E}}_{7}}$=E7+ ${\displaystyle {\bar {T}}_{8}}$=E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [[3<sup>3,3,1</sup>]] [34,3,1]
Order 48 720 23,040 2,903,040
Graph - -
Name 13,-1 130 131 132 133 134

#### Rectified 133 honeycomb

Rectified 133 honeycomb
(no image)
TypeUniform tessellation
Schläfli symbol{33,3,1}
Coxeter symbol0331
Coxeter-Dynkin diagram
or
7-face typeTrirectified 7-simplex
Rectified 1_32
6-face typesBirectified 6-simplex
Birectified 6-cube
Rectified 1_22
5-face typesRectified 5-simplex
Birectified 5-simplex
Birectified 5-orthoplex
4-face type5-cell
Rectified 5-cell
24-cell
Cell type{3,3}
{3,4}
Face type{3}
Vertex figure{}×{3,3}×{3,3}
Coxeter group${\displaystyle {\tilde {E}}_{7}}$, [[3,3<sup>3,3</sup>]]
Propertiesvertex-transitive, facet-transitive

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

## Notes

1. N.W. Johnson: Geometries and Transformations, (2018) 12.4: Euclidean Coxeter groups, p.294
2. http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Es7.html
3. The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin

## 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
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 345]
• Klitzing, Richard. "7D Heptacombs o3o3o3o3o3o3o *d3x - linoh".
• Klitzing, Richard. "7D Heptacombs o3o3o3x3o3o3o *d3o - rolinoh".
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$ ${\displaystyle {\tilde {C}}_{n-1}}$ ${\displaystyle {\tilde {B}}_{n-1}}$ ${\displaystyle {\tilde {D}}_{n-1}}$ ${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21