# 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${\tilde {E}}_{7}$ , [[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 ${\tilde {E}}_{7}$ group is related to the ${\tilde {F}}_{4}$ by a geometric folding, so this honeycomb can be projected into the 4-dimensional demitesseractic honeycomb.

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

## E7* lattice

${\tilde {E}}_{7}$ contains ${\tilde {A}}_{7}$ as a subgroup of index 144. 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) 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. 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 ${\tilde {E}}_{7}$ =E7+ ${\bar {T}}_{8}$ =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${\tilde {E}}_{7}$ , [[3,3<sup>3,3</sup>]]
Propertiesvertex-transitive, facet-transitive

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