# 2 31 polytope

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In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

## Contents

Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.

The rectified 231 is constructed by points at the mid-edges of the 231.

These polytopes are part of a family of 127 (or 27−1) 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: .

## 2_31 polytope

The 231 is composed of 126 vertices, 2016 edges, 10080 faces (Triangles), 20160 cells (tetrahedra), 16128 4-faces (3-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 221). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E7.

This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 331.

## Alternate names

• E. L. Elte named it V126 (for its 126 vertices) in his 1912 listing of semiregular polytopes.
• It was called 231 by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
• Pentacontihexa-pentacosiheptacontihexa-exon (Acronym laq) - 56-576 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 short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 321 polytope, .

Removing the node on the end of the 3-length branch leaves the 221. There are 56 of these facets. These facets are centered on the locations of the vertices of the 132 polytope, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 131, .

## Rectified 2_31 polytope

The rectified 231 is a rectification of the 231 polytope, creating new vertices on the center of edge of the 231.

## Alternate names

• Rectified pentacontihexa-pentacosiheptacontihexa-exon - as a rectified 56-576 facetted polyexon (acronym rolaq) (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 short branch leaves the rectified 6-simplex, .

Removing the node on the end of the 2-length branch leaves the, 6-demicube, .

Removing the node on the end of the 3-length branch leaves the rectified 221, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node.

## References

2 31 polytope Wikipedia

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