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: .

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

• E. L. Elte named it V₂₇ (for its 27 vertices) in his 1912 listing of semiregular polytopes.^[3]
• Icosihepta-heptacontidi-peton - 27-72 facetted polypeton (acronym jak) (Jonathan Bowers)^[4]

Coordinates

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

• (-2,0,0,0,-2,0,0,0)(0,-2,0,0,-2,0,0,0)(0,0,-2,0,-2,0,0,0)(0,0,0,-2,-2,0,0,0)(0,0,0,0,-2,0,0,-2)(0,0,0,0,0,-2,-2,0)
• ( 2,0,0,0,-2,0,0,0)(0, 2,0,0,-2,0,0,0)(0,0, 2,0,-2,0,0,0)(0,0,0, 2,-2,0,0,0)(0,0,0,0,-2,0,0, 2)
• (-1,-1,-1,-1,-1,-1,-1,-1)
• (-1,-1,-1, 1,-1,-1,-1, 1) (-1,-1, 1,-1,-1,-1,-1, 1) (-1,-1, 1, 1,-1,-1,-1,-1) (-1, 1,-1,-1,-1,-1,-1, 1) (-1, 1,-1, 1,-1,-1,-1,-1) (-1, 1, 1,-1,-1,-1,-1,-1) (1,-1,-1,-1,-1,-1,-1, 1) (1,-1, 1,-1,-1,-1,-1,-1) (1,-1,-1, 1,-1,-1,-1,-1) (1, 1,-1,-1,-1,-1,-1,-1)
• (-1, 1, 1, 1,-1,-1,-1, 1) (1,-1, 1, 1,-1,-1,-1, 1) (1, 1,-1, 1,-1,-1,-1, 1) (1, 1, 1,-1,-1,-1,-1, 1) (1, 1, 1, 1,-1,-1,-1,-1)

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

• Rectified icosihepta-heptacontidi-peton as a rectified 27-72 facetted polypeton (acronym rojak) (Jonathan Bowers)^[5]

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

• List of E6 polytopes

Notes

References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen 
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
  • (Paper 17) Coxeter, The Evolution of Coxeter-Dynkin diagrams, [Nieuw Archief voor Wiskunde 9 (1991) 233-248] See figure 1: (p. 232) (Node-edge graph of polytope)
• Richard Klitzing, 6D, uniform polytopes (polypeta) x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak