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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.

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.

Contents

2_21 polytopeEdit

221 polytope
Type Uniform 6-polytope
Family k21 polytope
Schläfli symbol {3,3,32,1}
Coxeter symbol 221
Coxeter-Dynkin diagram           or        
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], 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 contains the 1-skeleton of this polytope.

Alternate namesEdit

  • 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]

CoordinatesEdit

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)

ConstructionEdit

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 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 node and ringing the neighboring node. This makes 5-demicube (121 polytope),        .

ImagesEdit

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 foldingEdit

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:          .

Related complex polyhedraEdit

The regular complex polygon 3{3}3{3}3,      , in   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.

Related polytopesEdit

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
Space Finite Euclidean Hyperbolic
n 5 6 7 8
Coxeter
group
A5 E6  =E6+ E6++
Coxeter
diagram
                                   
Graph    
Name 220 221 222 223

Rectified 2_21 polytopeEdit

Rectified 221 polytope
Type Uniform 6-polytope
Schläfli symbol t1{3,3,32,1}
Coxeter symbol t1(221)
Coxeter-Dynkin diagram           or        
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], 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 namesEdit

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

ConstructionEdit

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{},        .

ImagesEdit

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 alsoEdit

NotesEdit

  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. doi:10.2307/2371466. JSTOR 2371466. 
  3. ^ Elte, 1912
  4. ^ Klitzing, (x3o3o3o3o *c3o - jak)
  5. ^ Klitzing, (o3x3o3o3o *c3o - rojak)

ReferencesEdit

  • 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)
  • Klitzing, Richard. "6D uniform polytopes (polypeta)".  x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak
Fundamental convex regular and uniform polytopes in dimensions 2–10
An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Triangle Square p-gon Hexagon Pentagon
Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
5-simplex 5-orthoplex5-cube 5-demicube
6-simplex 6-orthoplex6-cube 6-demicube 122221
7-simplex 7-orthoplex7-cube 7-demicube 132231321
8-simplex 8-orthoplex8-cube 8-demicube 142241421
9-simplex 9-orthoplex9-cube 9-demicube
10-simplex 10-orthoplex10-cube 10-demicube
n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds