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2 31 polytope

Uniform Polytope

Uniform Polytope

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 seven dimensions, made of uniform polytope facets and vertex figures, defined by all combinations of rings in this Coxeter-Dynkin diagram: .

231 polytope

Gosset 231 polytope
Type
Family
Schläfli symbol
Coxeter symbol
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Petrie polygon
Coxeter group
Properties

The 231 is composed of 126 vertices, 2016 edges, 10080 faces (triangles), 20160 cells (tetrahedra), 16128 4-faces (4-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.
  • It was called 231 by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
  • Pentacontahexa-pentacosiheptacontahexa-exon (Acronym: laq) - 56-576 facetted polyexon (Jonathan Bowers)

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.

E7width=70k-facefkf0f1f2f3colspan=2f4colspan=2f5colspan=2f6k-figuresNotesf0f1f2f3f4f5f6
D6( )12632240640160
A5A1{ }22016156020
A3A2A1{3}331008084
A3A2{3,3}464201601
A4A2{3,3,3}5101054032
A4A1510105*12096
D5A1{3,3,3,4}1040808016
A5{3,3,3,3}61520150
E6{3,3,32,1}272167201080216
A6{3,3,3,3,3}72135350

Images

E7E6 / F4B6 / A6A5D7 / B6D6 / B5D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3
[[File:up2 2 31 t0 E7.svg200px]][18][[File:up2 2 31 t0 E6.svg200px]][12][[File:up2 2 31 t0 A6.svg200px]][7x2]
[[File:up2 2 31 t0 A5.svg200px]][6][[File:up2 2 31 t0 D7.svg200px]][12/2][[File:up2 2 31 t0 D6.svg200px]][10]
[[File:up2 2 31 t0 D5.svg200px]][8][[File:up2 2 31 t0 D4.svg200px]][6][[File:up2 2 31 t0 D3.svg200px]][4]

Rectified 231 polytope

Rectified 231 polytope
Type
Family
Schläfli symbol
Coxeter symbol
Coxeter diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Petrie polygon
Coxeter group
Properties

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

Alternate names

  • Rectified pentacontahexa-pentacosiheptacontahexa-exon - as a rectified 56-576 facetted polyexon (Acronym: rolaq) (Jonathan Bowers)

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

E7E6 / F4B6 / A6A5D7 / B6D6 / B5D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3
[[File:up2 2 31 t1 E7.svg200px]][18][[File:up2 2 31 t1 E6.svg200px]][12][[File:up2 2 31 t1 A6.svg200px]][7x2]
[[File:up2 2 31 t1 A5.svg200px]][6][[File:up2 2 31 t1 D7.svg200px]][12/2][[File:up2 2 31 t1 D6.svg200px]][10]
[[File:up2 2 31 t1 D5.svg200px]][8][[File:up2 2 31 t1 D4.svg200px]][6][[File:up2 2 31 t1 D3.svg200px]][4]

Notes

References

  • 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, wiley.com,
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
  • x3o3o3o *c3o3o3o - laq, o3x3o3o *c3o3o3o - rolaq

References

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