321

231

132

Rectified 321

birectified 321

Rectified 231

Rectified 132
Orthogonal projections in E7 Coxeter plane

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

Its Coxeter symbol is 231, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.

The rectified 231 is constructed by points at the mid-edges of the 231.

These polytopes are part of a family of 127 (or 27−1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

2_31 polytope

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Gosset 231 polytope
Type Uniform 7-polytope
Family 2k1 polytope
Schläfli symbol {3,3,33,1}
Coxeter symbol 231
Coxeter diagram            
6-faces 632:
56 221 
576 {35} 
5-faces 4788:
756 211 
4032 {34} 
4-faces 16128:
4032 201 
12096 {33} 
Cells 20160 {32} 
Faces 10080 {3} 
Edges 2016
Vertices 126
Vertex figure 131
 
Petrie polygon Octadecagon
Coxeter group E7, [33,2,1]
Properties convex

The 231 is composed of 126 vertices, 2016 edges, 10080 faces (Triangles), 20160 cells (tetrahedra), 16128 4-faces (3-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 221). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E7.

This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 331.

Alternate names

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  • E. L. Elte named it V126 (for its 126 vertices) in his 1912 listing of semiregular polytopes.[1]
  • It was called 231 by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
  • Pentacontihexa-pentacosiheptacontihexa-exon (Acronym laq) - 56-576 facetted polyexon (Jonathan Bowers)[2]

Construction

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It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.

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

Removing the node on the short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 321 polytope,            .

Removing the node on the end of the 3-length branch leaves the 221. There are 56 of these facets. These facets are centered on the locations of the vertices of the 132 polytope,          .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 131,          .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.[3]

E7             k-face fk f0 f1 f2 f3 f4 f5 f6 k-figures notes
D6             ( ) f0 126 32 240 640 160 480 60 192 12 32 6-demicube E7/D6 = 72x8!/32/6! = 126
A5A1             { } f1 2 2016 15 60 20 60 15 30 6 6 rectified 5-simplex E7/A5A1 = 72x8!/6!/2 = 2016
A3A2A1             {3} f2 3 3 10080 8 4 12 6 8 4 2 tetrahedral prism E7/A3A2A1 = 72x8!/4!/3!/2 = 10080
A3A2             {3,3} f3 4 6 4 20160 1 3 3 3 3 1 tetrahedron E7/A3A2 = 72x8!/4!/3! = 20160
A4A2             {3,3,3} f4 5 10 10 5 4032 * 3 0 3 0 {3} E7/A4A2 = 72x8!/5!/3! = 4032
A4A1             5 10 10 5 * 12096 1 2 2 1 Isosceles triangle E7/A4A1 = 72x8!/5!/2 = 12096
D5A1             {3,3,3,4} f5 10 40 80 80 16 16 756 * 2 0 { } E7/D5A1 = 72x8!/32/5! = 756
A5             {3,3,3,3} 6 15 20 15 0 6 * 4032 1 1 E7/A5 = 72x8!/6! = 72*8*7 = 4032
E6             {3,3,32,1} f6 27 216 720 1080 216 432 27 72 56 * ( ) E7/E6 = 72x8!/72x6! = 8*7 = 56
A6             {3,3,3,3,3} 7 21 35 35 0 21 0 7 * 576 E7/A6 = 72x8!/7! = 72×8 = 576

Images

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Coxeter plane projections
E7 E6 / F4 B6 / A6
 
[18]
 
[12]
 
[7x2]
A5 D7 / B6 D6 / B5
 
[6]
 
[12/2]
 
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
 
[8]
 
[6]
 
[4]
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2k1 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 =   = E8+ E10 =   = E8++
Coxeter
diagram
                                                                                         
Symmetry [3−1,2,1] [30,2,1] [[31,2,1]] [32,2,1] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 384 51,840 2,903,040 696,729,600
Graph             - -
Name 2−1,1 201 211 221 231 241 251 261

Rectified 2_31 polytope

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Rectified 231 polytope
Type Uniform 7-polytope
Family 2k1 polytope
Schläfli symbol {3,3,33,1}
Coxeter symbol t1(231)
Coxeter diagram            
6-faces 758
5-faces 10332
4-faces 47880
Cells 100800
Faces 90720
Edges 30240
Vertices 2016
Vertex figure 6-demicube
Petrie polygon Octadecagon
Coxeter group E7, [33,2,1]
Properties convex

The rectified 231 is a rectification of the 231 polytope, creating new vertices on the center of edge of the 231.

Alternate names

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  • Rectified pentacontihexa-pentacosiheptacontihexa-exon - as a rectified 56-576 facetted polyexon (acronym rolaq) (Jonathan Bowers)[4]

Construction

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It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.

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

Removing the node on the short branch leaves the rectified 6-simplex,            .

Removing the node on the end of the 2-length branch leaves the, 6-demicube,          .

Removing the node on the end of the 3-length branch leaves the rectified 221,          .

The vertex figure is determined by removing the ringed node and ringing the neighboring node.

         

Images

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Coxeter plane projections
E7 E6 / F4 B6 / A6
 
[18]
 
[12]
 
[7x2]
A5 D7 / B6 D6 / B5
 
[6]
 
[12/2]
 
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
 
[8]
 
[6]
 
[4]

See also

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Notes

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  1. ^ Elte, 1912
  2. ^ Klitzing, (x3o3o3o *c3o3o3o - laq)
  3. ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
  4. ^ Klitzing, (o3x3o3o *c3o3o3o - rolaq)

References

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  • Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
  • H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • 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 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3o *c3o3o3o - laq, o3x3o3o *c3o3o3o - rolaq
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 polychoron Pentachoron 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