2 31 polytope

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


321

231

132

Rectified 321

birectified 321

Rectified 231

Rectified 132
Orthogonal projections in E7 Coxeter plane

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 271) 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

Gosset 231 polytope
TypeUniform 7-polytope
Family2k1 polytope
Schläfli symbol{3,3,33,1}
Coxeter symbol231
Coxeter diagram
6-faces632:
56 221
576 {35}
5-faces4788:
756 211
4032 {34}
4-faces16128:
4032 201
12096 {33}
Cells20160 {32}
Faces10080 {3}
Edges2016
Vertices126
Vertex figure131
Petrie polygonOctadecagon
Coxeter groupE7, [33,2,1]
Propertiesconvex

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

  • 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

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]

E7k-facefkf0f1f2f3f4f5f6k-figuresnotes
D6( ) f0 126322406401604806019212326-demicubeE7/D6 = 72x8!/32/6! = 126
A5A1{ } f1 2201615602060153066rectified 5-simplexE7/A5A1 = 72x8!/6!/2 = 2016
A3A2A1{3} f2 331008084126842tetrahedral prismE7/A3A2A1 = 72x8!/4!/3!/2 = 10080
A3A2 {3,3} f3 46420160133331tetrahedronE7/A3A2 = 72x8!/4!/3! = 20160
A4A2 {3,3,3} f4 5101054032*3030{3}E7/A4A2 = 72x8!/5!/3! = 4032
A4A1 510105*120961221Isosceles triangleE7/A4A1 = 72x8!/5!/2 = 12096
D5A1 {3,3,3,4} f5 104080801616756*20{ }E7/D5A1 = 72x8!/32/5! = 756
A5 {3,3,3,3} 615201506*403211E7/A5 = 72x8!/6! = 72*8*7 = 4032
E6 {3,3,32,1} f6 272167201080216432277256*( )E7/E6 = 72x8!/72x6! = 8*7 = 56
A6 {3,3,3,3,3} 721353502107*576E7/A6 = 72x8!/7! = 72×8 = 576

Images

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]

Rectified 2_31 polytope

Rectified 231 polytope
TypeUniform 7-polytope
Family2k1 polytope
Schläfli symbol{3,3,33,1}
Coxeter symbolt1(231)
Coxeter diagram
6-faces758
5-faces10332
4-faces47880
Cells100800
Faces90720
Edges30240
Vertices2016
Vertex figure6-demicube
Petrie polygonOctadecagon
Coxeter groupE7, [33,2,1]
Propertiesconvex

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

Alternate names

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

Construction

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

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

Notes

  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

  • 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
    • (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 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
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