1 32 polytope


321
Rectified 321
Birectified 321
Rectified 132

132
231
Rectified 231
Orthogonal projections in E6 Coxeter plane

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

Coxeter named it 132 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.

The rectified 132 is constructed by points at the mid-edges of the 132.

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

1_32 polytope

132
Type Uniform 7-polytope
Family 1k2 polytope
Schläfli symbol {3,33,2}
Coxeter symbol 132
Coxeter-Dynkin diagram
6-faces 182:
56 122
126 131
5-faces 4284:
756 121
1512 121
2016 {34}
4-faces 23688:
4032 {33}
7560 111
12096 {33}
Cells 50400:
20160 {32}
30240 {32}
Faces 40320 {3}
Edges 10080
Vertices 576
Vertex figure t2{35}
Petrie polygon Octadecagon
Coxeter group E7, [33,2,1]
Properties convex

This polytope can tessellate 7-dimensional space, with symbol 133, and Coxeter-Dynkin diagram, .

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 end of the 2-length branch leaves the 6-demicube, 131,

Removing the node on the end of the 3-length branch leaves the 122,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 032,

Images

Coxeter plane projections
E7 E6 / F4 B7 / 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 1_32 polytope

Rectified 132
Type Uniform 7-polytope
Schläfli symbol t1{3,33,2}
Coxeter symbol t1(132)
Coxeter-Dynkin diagram
6-faces 758
5-faces 12348
4-faces 72072
Cells 191520
Faces 241920
Edges 120960
Vertices 10080
Vertex figure {3,3}×{3}×{}
Coxeter group E7, [33,2,1]
Properties convex

The rectified 132 is a rectification of the 132 polytope, creating new vertices on the center of edge of the 132. 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 122 polytope,

Removing the node on the end of the 2-length branch leaves the demihexeract, 131,

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}×{},

Images

Coxeter plane projections
E7 E6 / F4 B7 / 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, (o3o3o3x *c3o3o3o - lin)
  3. ^ Klitzing, (o3o3x3o *c3o3o3o - rolin)

References