2 21 polytope


221
Rectified 221

(122)
Birectified 221
(Rectified 122)
orthogonal projections in E6 Coxeter plane

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 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 221 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied[2] its connection with the 27 lines on the cubic surface, which are naturally in correspondence with the vertices of 221.

The rectified 221 is constructed by points at the mid-edges of the 221. The birectified 221 is constructed by points at the triangle face centers of the 221, and is the same as the rectified 122.

These polytopes are a part of family of 39 convex uniform polytopes in 6-dimensions, made of uniform 5-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

Contents

2_21 polytope

221 polytope
Type Uniform 6-polytope
Family k21 polytope
Schläfli symbol {3,3,32,1}
Coxeter symbol 221
Coxeter-Dynkin diagram
5-faces 99 total:
27 211
72 {34}
4-faces 648:
432 {33}
216 {33}
Cells 1080 {3,3}
Faces 720 {3}
Edges 216
Vertices 27
Vertex figure 121 (5-demicube)
Petrie polygon Dodecagon
Coxeter group E6, [32,2,1]
Properties convex

The 221 has 27 vertices, and 99 facets: 27 5-orthoplexes and 72 5-simplices. Its vertex figure is a 5-demicube.

For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon (called a Petrie polygon). Its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements (faces, cells, etc) can also be extracted and drawn on this projection.

Alternate names

Coordinates

The 27 vertices can be expressed in 8-space as an edge-figure of the 421 polytope:

Construction

Its construction is based on the E6 group.

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

Removing the ring on the short branch leaves the 5-simplex, .

Removing the ring on the end of the 2-length branch leaves the 5-orthoplex in its alternated form: (211), .

Every simplex facet touches an 5-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure is determined by removing the ringed ring and ringing the neighboring ring. This makes 5-demicube (121 polytope), .

Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow. The number of vertices by color are given in parentheses.

Coxeter plane orthographic projections
E6
[12]		 D5
[8]		 D4 / A2
[6]		 B6
[12/2]

(1,3)
(1,3)
(3,9)
(1,3)
A5
[6]		 A4
[5]		 A3 / D3
[4]

(1,3)
(1,2)
(1,4,7)

Geometric folding

The 221 is related to the 24-cell by a geometric folding of the E6/F4 Coxeter-Dynkin diagrams. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 221.

E6
 F4

221
24-cell

This polytope can tessellate Euclidean 6-space, forming the 222 honeycomb with this Coxeter-Dynkin diagram: .

Rectified 2_21 polytope

Rectified 221 polytope
Type Uniform 6-polytope
Schläfli symbol t1{3,3,32,1}
Coxeter symbol t1(221)
Coxeter-Dynkin diagram
5-faces 126 total:

72 t1{34}
27 t1{33,4}
27 t1{3,32,1}
4-faces 1350
Cells 4320
Faces 5040
Edges 2160
Vertices 216
Vertex figure rectified 5-cell prism
Coxeter group E6, [32,2,1]
Properties convex

The rectified 221 has 216 vertices, and 126 facets: 72 rectified 5-simplices, and 27 rectified 5-orthoplexes and 27 5-demicubes . Its vertex figure is a rectified 5-cell prism.

Alternate names

Construction

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

Removing the ring on the short branch leaves the rectified 5-simplex, .

Removing the ring on the end of the other 2-length branch leaves the rectified 5-orthoplex in its alternated form: t1(211), .

Removing the ring on the end of the same 2-length branch leaves the 5-demicube: (121), .

The vertex figure is determined by removing the ringed ring and ringing the neighboring ring. This makes rectified 5-cell prism, t1{3,3,3}x{}, .

Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Coxeter plane orthographic projections
E6
[12]		 D5
[8]		 D4 / A2
[6]		 B6
[12/2]
A5
[6]		 A4
[5]		 A3 / D3
[4]

See also

Notes

  1. ^ Gosset, 1900
  2. ^ Coxeter, H.S.M. (1940). "The Polytope 221 Whose Twenty-Seven Vertices Correspond to the Lines on the General Cubic Surface". Amer. J. Math. 62: 457–486. http://www.jstor.org/stable/2371466. 
  3. ^ Elte, 1912
  4. ^ Klitzing, (x3o3o3o3o *c3o - jak)
  5. ^ Klitzing, (o3x3o3o3o *c3o - rojak)

References