2 31 polytope


321
Rectified 321
Birectified 321
Rectified 132

132
231
Rectified 231
Orthogonal projections in E6 Coxeter plane

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

Coxeter named it 231 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences.

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

These polytopes are part of a family of 127 (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: .

Contents

2_31 polytope

Gosset 231 polytope
Type Uniform 7-polytope
Family 2k1 polytope
Schläfli symbol {3,3,33,1}
Coxeter symbol 231
Coxeter-Dynkin diagram
6-faces 632:
56 221
576 {35}
5-faces 4788:
756 211
4032 {34}
4-faces 16128:
4032 201
12096 {33}
Cells 20160 {32}
Faces 10080 {3}
Edges 2016
Vertices 126
Vertex figure 131
Petrie polygon Octadecagon
Coxeter group E7, [33,2,1]
Properties convex

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

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 56 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 576 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, .

Images

Coxeter plane projections
E7 E6 / F4 B6 / A6

[18]
[12]
[7x2]
A5 D7 / B6 D6 / B5

[6]
[12/2]
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3

[8]
[6]
[4]

Rectified 2_31 polytope

Rectified 231 polytope
Type Uniform 7-polytope
Family 2k1 polytope
Schläfli symbol {3,3,33,1}
Coxeter symbol 231
Coxeter-Dynkin diagram
6-faces 758
5-faces 10332
4-faces 47880
Cells 100800
Faces 90720
Edges 30240
Vertices 2016
Vertex figure 6-demicube
Petrie polygon Octadecagon
Coxeter group E7, [33,2,1]
Properties convex

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

Alternate names

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. This makes the 6-demicube, 131, .

Images

Coxeter plane projections
E7 E6 / F4 B6 / A6

[18]
[12]
[7x2]
A5 D7 / B6 D6 / B5

[6]
[12/2]
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3

[8]
[6]
[4]

See also

Notes

  1. ^ Elte, 1912
  2. ^ Klitzing, (x3o3o3o *c3o3o3o - laq)
  3. ^ Klitzing, (o3x3o3o *c3o3o3o - rolaq)

References