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 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]
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

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 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
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