# 2 21 polytope

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

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] It is also called the Schläfli polytope.

## Contents

Its Coxeter symbol is 221, describing 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 symbol221
Coxeter-Dynkin diagram or
5-faces99 total:
27 211
72 {34}
4-faces648:
432 {33}
216 {33}
Cells1080 {3,3}
Faces720 {3}
Edges216
Vertices27
Vertex figure 121 (5-demicube)
Petrie polygon Dodecagon
Coxeter group E6, [32,2,1], order 51840
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.

The Schläfli graph is the 1-skeleton of this polytope.

### 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 node on the short branch leaves the 5-simplex, .

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

Every simplex facet touches a 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 node and ringing the neighboring node. This makes 5-demicube (121 polytope), . The edge-figure is the vertex figure of the vertex figure, a rectified 5-cell, (021 polytope), .

Seen in a configuration matrix, the element counts can be derived from the Coxeter group orders. [5]

E6k-facefkf0f1f2f3f4f5k-figurenotes
D5( )f027168016080401610 h{4,3,3,3} E6/D5 = 51840/1920 = 27
A4A1{ }f122161030201055 r{3,3,3} E6/A4A1 = 51840/120/2 = 216
A2A2A1 {3} f23372066323 {3}x{ } E6/A2A2A1 = 51840/6/6/2 = 720
A3A1 {3,3} f346410802112 { }v( ) E6/A3A1 = 51840/24/2 = 1080
A4 {3,3,3} f4510105432*11{ }E6/A4 = 51840/120 = 432
A4A1510105*21602E6/A4A1 = 51840/120/2 = 216
A5 {3,3,3,3} f561520156072*( )E6/A5 = 51840/720 = 72
D5 {3,3,3,4} 104080801616*27E6/D5 = 51840/1920 = 27

### 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: .

The regular complex polygon 3{3}3{3}3, , in ${\displaystyle \mathbb {C} ^{2}}$ has a real representation as the 221 polytope, , in 4-dimensional space. It is called a Hessian polyhedron after Edmund Hess. It has 27 vertices, 72 3-edges, and 27 3{3}3 faces. Its complex reflection group is 3[3]3[3]3, order 648.

The 221 is fourth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.

The 221 polytope is fourth in dimensional series 2k2.

The 221 polytope is second in dimensional series 22k.

22k figures of n dimensions
SpaceFiniteEuclideanHyperbolic
n 4 5 6 7 8
Coxeter
group
A2A2A5E6${\displaystyle {\tilde {E}}_{6}}$=E6+E6++
Coxeter
diagram
Graph
Name 22,-1 220 221 222 223

## Rectified 2_21 polytope

Rectified 221 polytope
Type Uniform 6-polytope
Schläfli symbol t1{3,3,32,1}
Coxeter symbolt1(221)
Coxeter-Dynkin diagram or
5-faces126 total:

72 t1{34}
27 t1{33,4}
27 t1{3,32,1}

4-faces1350
Cells4320
Faces5040
Edges2160
Vertices216
Vertex figure rectified 5-cell prism
Coxeter group E6, [32,2,1], order 51840
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) [6]

### 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]

## Truncated 2_21 polytope

Truncated 221 polytope
Type Uniform 6-polytope
Schläfli symbol t{3,3,32,1}
Coxeter symbolt(221)
Coxeter-Dynkin diagram or
5-faces72+27+27
4-faces432+216+432+270
Cells1080+2160+1080
Faces720+4320
Edges216+2160
Vertices432
Vertex figure ( ) v r{3,3,3}
Coxeter group E6, [32,2,1], order 51840
Properties convex

The truncated 221 has 432 vertices, 5040 edges, 4320 faces, 1350 cells, and 126 4-faces. Its vertex figure is a rectified 5-cell pyramid.

### Images

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

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

## 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 (1): 457–486. doi:10.2307/2371466. JSTOR   2371466.
3. Elte, 1912
4. Klitzing, (x3o3o3o3o *c3o - jak)
5. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
6. Klitzing, (o3x3o3o3o *c3o - rojak)

## Related Research Articles

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In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence.

In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge.

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

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).

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

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

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

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

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In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E10) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded.

## 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
• (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)
• Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds