In 7-dimensional geometry, **1 _{32}** is a uniform polytope, constructed from the E7 group.

## Contents

- 132 polytope
- Alternate names
- Construction
- Related polytopes and honeycombs
- Rectified 132 polytope
- References

Its Coxeter symbol is **1 _{32}**, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.

The **rectified 1 _{32}** is constructed by points at the mid-edges of the

**1**.

_{32}These polytopes are part of a family of 127 (2^{7}-1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

## 1_32 polytope

This polytope can tessellate 7-dimensional space, with symbol **1 _{33}**, and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual E

_{7}

^{*}lattice.

## Alternate names

_{576}(for its 576 vertices) in his 1912 listing of semiregular polytopes.

**1**for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.

_{32}*Pentacontihexa-hecatonicosihexa-exon*(Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers)

## Construction

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram,

Removing the node on the end of the 2-length branch leaves the 6-demicube, 1_{31},

Removing the node on the end of the 3-length branch leaves the 1_{22},

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 0_{32},

## Related polytopes and honeycombs

The 1_{32} is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. The next figure is the Euclidean honeycomb 1_{33} and the final is a noncompact hyperbolic honeycomb, 1_{34}.

## Rectified 1_32 polytope

The **rectified 1 _{32}** (also called

**0**) is a rectification of the 1

_{321}_{32}polytope, creating new vertices on the center of edge of the 1

_{32}. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}.

## Alternate names

## Construction

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, , and the ring represents the position of the active mirror(s).

Removing the node on the end of the 3-length branch leaves the rectified 1_{22} polytope,

Removing the node on the end of the 2-length branch leaves the demihexeract, 1_{31},

Removing the node on the end of the 1-length branch leaves the birectified 6-simplex,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×{},