3 21 polytope


231
Rectified 321
Birectified 321
Rectified 132

132
231
Rectified 231
Orthogonal projections in E6 Coxeter plane

In 7-dimensional geometry, the 321 polytope is a uniform 6-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure.[1]

Coxeter named it 321 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.

The rectified 321 is constructed by points at the mid-edges of the 321. The birectified 321 is constructed by points at the triangle face centers of the 321. The trirectified 321 is constructed by points at the tetrahedral centers of the 321, and is the same as the rectified 132.

These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

Contents

321 polytope

This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 331 and Coxeter-Dynkin diagram: .

321 polytope
Type Uniform 7-polytope
Family k21 polytope
Schläfli symbol {3,3,3,32,1}
Coxeter symbol 321
Coxeter-Dynkin diagram
6-faces 702 total:
126 311
576 {35}
5-faces 6048:
4032 {34}
2016 {34}
4-faces 12096 {33}
Cells 10080 {3,3}
Faces 4032 {3}
Edges 756
Vertices 56
Vertex figure 221 polytope
Petrie polygon octadecagon
Coxeter group E7, [33,2,1]
Properties convex

In 7-dimensional geometry, the 321 is a uniform polytope. It has 56 vertices, and 702 facets: 126 311 and 576 6-simplex.

For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within a 18-gonal regular polygon (called a Petrie polygon). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc) can also be extracted and drawn on this projection.

The 1-skeleton of the 321 polytope is called a Gosset graph.

Alternate names

Coordinates

The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite:

± (-3, -3, 1, 1, 1, 1, 1, 1)

Construction

Its construction is based on the E7 group. Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single nodea_1 on the end of the 3-node sequence.

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

Removing the node on the short branch leaves the 6-simplex, .

Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its alternated form: 311, .

Every simplex facet touches an 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 221 polytope, .

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 321 polytope

Rectified 321 polytope
Type Uniform 7-polytope
Schläfli symbol t1{3,3,3,32,1}
Coxeter symbol t1(321)
Coxeter-Dynkin diagram
6-faces 758
5-faces 44352
4-faces 70560
Cells 48384
Faces 11592
Edges 12096
Vertices 756
Vertex figure 5-demicube prism
Petrie polygon octadecagon
Coxeter group E7, [33,2,1]
Properties convex

Alternate names

Construction

Its construction is based on the E7 group. Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.

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

Removing the node on the short branch leaves the 6-simplex, .

Removing the node on the end of the 2-length branch leaves the rectified 6-orthoplex in its alternated form: t1311, .

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

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 5-demicube prism, .

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]

Birectified 321 polytope

Birectified 321 polytope
Type Uniform 7-polytope
Schläfli symbol t1{3,3,3,32,1}
Coxeter symbol t1(321)
Coxeter-Dynkin diagram
6-faces 758
5-faces 12348
4-faces 68040
Cells 161280
Faces 161280
Edges 60480
Vertices 4032
Vertex figure 5-cell-triangle duoprism
Petrie polygon octadecagon
Coxeter group E7, [33,2,1]
Properties convex

Alternate names

Construction

Its construction is based on the E7 group. Coxeter named it as 321 by its bifurcating Coxeter-Dynkin diagram, with a single node on the end of the 3-node sequence.

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

Removing the node on the short branch leaves the birectified 6-simplex, .

Removing the node on the end of the 2-length branch leaves the birectified 6-orthoplex in its alternated form: t2(311), .

Removing the node on the end of the 3-length branch leaves the rectified 221 polytope in its alternated form: t1(221), .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 5-cell-triangle duoprism, .

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. ^ a b Gosset, 1900
  2. ^ Elte, 1912
  3. ^ Klitzing, (o3o3o3o *c3o3o3x - naq)
  4. ^ Klitzing. (o3o3o3o *c3o3x3o - ranq)
  5. ^ Klitzing, (o3o3o3o *c3x3o3o - branq)

References