In 7-dimensional geometry, **1 _{33}** is a uniform honeycomb, also given by Schläfli symbol {3,3

^{3,3}}, and is composed of 1

_{32}facets.

## Contents

## 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 1_{32}, 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, 0_{33}.

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

## E7* lattice

The **E _{7}^{*} lattice** (also called E

_{7}

^{2}) has double the symmetry, represented by [[3,3

^{3,3}]]. The Voronoi cell of the E

_{7}

^{*}lattice is the 1

_{32}polytope, and voronoi tessellation the

**1**. The

_{33}honeycomb**E**is constructed by 2 copies of the E

_{7}^{*}lattice_{7}lattice vertices, one from each long branch of the Coxeter diagram, and can be constructed as the union of four A

_{7}

^{*}lattices, also called A

_{7}

^{4}:

## Related polytopes and honeycombs

The 1_{33} is fourth in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. The final is a noncompact hyperbolic honeycomb, 1_{34}.

## Rectified 1_33 honeycomb

The *rectified 1 _{33}* or

*0*, Coxeter diagram has facets and , and vertex figure .

_{331}