2 21 polytope

Last updated
Up 2 21 t0 E6.svg
221
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 2 21 t1 E6.svg
Rectified 221
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 1 22 t0 E6.svg
(122)
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up 2 21 t2 E6.svg
Birectified 221
(Rectified 122)
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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] 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: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

2_21 polytope

221 polytope
Type Uniform 6-polytope
Family k21 polytope
Schläfli symbol {3,3,32,1}
Coxeter symbol221
Coxeter-Dynkin diagram CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png or CDel nodes 10r.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
5-faces99 total:
27 211 5-orthoplex.svg
72 {34} 5-simplex t0.svg
4-faces648:
432 {33} 4-simplex t0.svg
216 {33} 4-simplex t0.svg
Cells1080 {3,3} 3-simplex t0.svg
Faces720 {3} 2-simplex t0.svg
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

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, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

Removing the node on the short branch leaves the 5-simplex, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form: (211), CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png.

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), CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png. The edge-figure is the vertex figure of the vertex figure, a rectified 5-cell, (021 polytope), CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.png.

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

E6CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngk-facefkf0f1f2f3f4f5k-figurenotes
D5CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png( )f027168016080401610 h{4,3,3,3} E6/D5 = 51840/1920 = 27
A4A1CDel nodea 1.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png{ }f122161030201055 r{3,3,3} E6/A4A1 = 51840/120/2 = 216
A2A2A1CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x0.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png {3} f23372066323 {3}x{ } E6/A2A2A1 = 51840/6/6/2 = 720
A3A1CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png {3,3} f346410802112 { }v( ) E6/A3A1 = 51840/24/2 = 1080
A4CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png {3,3,3} f4510105432*11{ }E6/A4 = 51840/120 = 432
A4A1CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png510105*21602E6/A4A1 = 51840/120/2 = 216
A5CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png {3,3,3,3} f561520156072*( )E6/A5 = 51840/720 = 72
D5CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png {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]
Up 2 21 t0 E6.svg
(1,3)
Up 2 21 t0 D5.svg
(1,3)
Up 2 21 t0 D4.svg
(3,9)
Up 2 21 t0 B6.svg
(1,3)
A5
[6]
A4
[5]
A3 / D3
[4]
Up 2 21 t0 A5.svg
(1,3)
Up 2 21 t0 A4.svg
(1,2)
Up 2 21 t0 D3.svg
(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
Dyn-node.png Dyn-3.png Dyn-loop1.png Dyn-nodes.png Dyn-3s.png Dyn-nodes.png
F4
Dyn2-node.png Dyn2-3.png Dyn2-node.png Dyn2-4b.png Dyn2-node.png Dyn2-3.png Dyn2-node.png
E6 graph.svg
221
CDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes 10l.png
24-cell t3 F4.svg
24-cell
CDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png

This polytope can tessellate Euclidean 6-space, forming the 222 honeycomb with this Coxeter-Dynkin diagram: CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.png.

The regular complex polygon 3{3}3{3}3, CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png, in has a real representation as the 221 polytope, CDel nodes 10r.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png, 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=E6+E6++
Coxeter
diagram
CDel nodes 10r.pngCDel 3ab.pngCDel nodes.pngCDel nodes 10r.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel nodes 10r.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodes 10r.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel nodes 10r.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Graph 5-simplex t0.svg Up 2 21 t0 E6.svg
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 CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png or CDel nodes.pngCDel 3ab.pngCDel nodes 10lru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
5-faces126 total:

72 t1{34} 5-simplex t1.svg
27 t1{33,4} 5-cube t3.svg
27 t1{3,32,1} 5-demicube t0 D5.svg

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

Construction

Its construction is based on the E6 group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png.

Removing the ring on the short branch leaves the rectified 5-simplex, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png.

Removing the ring on the end of the other 2-length branch leaves the rectified 5-orthoplex in its alternated form: t1(211), CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png.

Removing the ring on the end of the same 2-length branch leaves the 5-demicube: (121), CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png.

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{}, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 2.pngCDel nodea 1.png.

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]
Up 2 21 t1 E6.svg Up 2 21 t1 D5.svg Up 2 21 t1 D4.svg Up 2 21 t1 B6.svg
A5
[6]
A4
[5]
A3 / D3
[4]
Up 2 21 t1 A5.svg Up 2 21 t1 A4.svg Up 2 21 t1 D3.svg

Truncated 2_21 polytope

Truncated 221 polytope
Type Uniform 6-polytope
Schläfli symbol t{3,3,32,1}
Coxeter symbolt(221)
Coxeter-Dynkin diagram CDel nodea 1.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png or CDel nodes 10r.pngCDel 3ab.pngCDel nodes 10lru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
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]
Up 2 21 t01 E6.svg Up 2 21 t01 D5.svg Up 2 21 t01 D4.svg Up 2 21 t01 B6.svg
A5
[6]
A4
[5]
A3 / D3
[4]
Up 2 21 t01 A5.svg Up 2 21 t01 A4.svg Up 2 21 t01 D3.svg

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 (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)

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Gosset–Elte figures

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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.

Uniform 10-polytope

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Rectified 5-simplexes

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2<sub> 31</sub> polytope

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

1<sub> 22</sub> polytope

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1<sub> 32</sub> polytope

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

1<sub> 42</sub> polytope

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

2<sub> 41</sub> polytope

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

3<sub> 21</sub> polytope

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.

4<sub> 21</sub> polytope

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.

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Rectified 6-simplexes

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

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.

Rectified 8-simplexes

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

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 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