2 31 polytope

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Up2 3 21 t0 E7.svg
321
CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 2 31 t0 E7.svg
231
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Up2 1 32 t0 E7.svg
132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 3 21 t1 E7.svg
Rectified 321
CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 3 21 t2 E7.svg
birectified 321
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Up2 2 31 t1 E7.svg
Rectified 231
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.png
Up2 1 32 t1 E7.svg
Rectified 132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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: CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

2_31 polytope

Gosset 231 polytope
Type Uniform 7-polytope
Family 2k1 polytope
Schläfli symbol {3,3,33,1}
Coxeter symbol 231
Coxeter diagram CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
6-faces 632:
56 221E6 graph.svg
576 {35}6-simplex t0.svg
5-faces 4788:
756 2115-orthoplex.svg
4032 {34}5-simplex t0.svg
4-faces 16128:
4032 2014-simplex t0.svg
12096 {33}4-simplex t0.svg
Cells 20160 {32}3-simplex t0.svg
Faces 10080 {3}2-simplex t0.svg
Edges 2016
Vertices 126
Vertex figure 131
6-demicube.svg
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

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

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

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

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 131, CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

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

E7 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png k-face fk f0 f1 f2 f3 f4 f5 f6 k-figures notes
D6 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png ( ) f0 126 32 240 640 160 480 60 192 12 32 6-demicube E7/D6 = 72x8!/32/6! = 126
A5A1 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea 1.png { } f1 2 2016 15 60 20 60 15 30 6 6 rectified 5-simplex E7/A5A1 = 72x8!/6!/2 = 2016
A3A2A1 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodes x0.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3} f2 3 3 10080 8 4 12 6 8 4 2 tetrahedral prism E7/A3A2A1 = 72x8!/4!/3!/2 = 10080
A3A2 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3} f3 4 6 4 20160 1 3 3 3 3 1 tetrahedron E7/A3A2 = 72x8!/4!/3! = 20160
A4A2 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,3} f4 5 10 10 5 4032 * 3 0 3 0 {3} E7/A4A2 = 72x8!/5!/3! = 4032
A4A1 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png 5 10 10 5 * 12096 1 2 2 1 Isosceles triangle E7/A4A1 = 72x8!/5!/2 = 12096
D5A1 CDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,3,4} f5 10 40 80 80 16 16 756 * 2 0 { } E7/D5A1 = 72x8!/32/5! = 756
A5 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,3,3} 6 15 20 15 0 6 * 4032 1 1 E7/A5 = 72x8!/6! = 72*8*7 = 4032
E6 CDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,32,1} f6 27 216 720 1080 216 432 27 72 56 * ( ) E7/E6 = 72x8!/72x6! = 8*7 = 56
A6 CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png {3,3,3,3,3} 7 21 35 35 0 21 0 7 * 576 E7/A6 = 72x8!/7! = 72×8 = 576

Images

Coxeter plane projections
E7 E6 / F4 B6 / A6
Up2 2 31 t0 E7.svg
[18]
Up2 2 31 t0 E6.svg
[12]
Up2 2 31 t0 A6.svg
[7x2]
A5 D7 / B6 D6 / B5
Up2 2 31 t0 A5.svg
[6]
Up2 2 31 t0 D7.svg
[12/2]
Up2 2 31 t0 D6.svg
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
Up2 2 31 t0 D5.svg
[8]
Up2 2 31 t0 D4.svg
[6]
Up2 2 31 t0 D3.svg
[4]

Related polytopes and honeycombs

Rectified 2_31 polytope

Rectified 231 polytope
Type Uniform 7-polytope
Family 2k1 polytope
Schläfli symbol {3,3,33,1}
Coxeter symbol t1(231)
Coxeter diagram CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
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

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

Removing the node on the short branch leaves the rectified 6-simplex, CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 2-length branch leaves the, 6-demicube, CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

Removing the node on the end of the 3-length branch leaves the rectified 221, CDel nodea.pngCDel 3a.pngCDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png.

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

CDel nodea 1.pngCDel 2.pngCDel branch 10.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Images

Coxeter plane projections
E7 E6 / F4 B6 / A6
Up2 2 31 t1 E7.svg
[18]
Up2 2 31 t1 E6.svg
[12]
Up2 2 31 t1 A6.svg
[7x2]
A5 D7 / B6 D6 / B5
Up2 2 31 t1 A5.svg
[6]
Up2 2 31 t1 D7.svg
[12/2]
Up2 2 31 t1 D6.svg
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
Up2 2 31 t1 D5.svg
[8]
Up2 2 31 t1 D4.svg
[6]
Up2 2 31 t1 D3.svg
[4]

See also

  • List of E7 polytopes

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 [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)". https://bendwavy.org/klitzing/dimensions/polyexa.htm.  x3o3o3o *c3o3o3o - laq, o3x3o3o *c3o3o3o - rolaq
Fundamental convex regular and uniform polytopes in dimensions 2–10
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