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1 32 polytope

Uniform polytope

Summary

Uniform polytope

Orthogonal projections in E7 Coxeter plane

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.

The rectified 132 is constructed by points at the mid-edges of the 132.

These polytopes are part of a family of 127 (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: .

132 polytope

132
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

This polytope can tessellate 7-dimensional space, with symbol 133, and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual E7* lattice.

Alternate names

  • Emanuel Lodewijk Elte named it V576 (for its 576 vertices) in his 1912 listing of semiregular polytopes.
  • Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
  • Pentacontahexa-hecatonicosihexa-exon (Acronym: lin) - 56-126 facetted polyexon (Jonathan Bowers)

Images

E7E6 / F4B7 / A6A5D7 / B6D6 / B5D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3
[[File:up2 1 32 t0 E7.svg200px]][18][[File:up2 1 32 t0 E6.svg200px]][12][[File:up2 1 32 t0 A6.svg200px]][7x2]
[[File:up2 1 32 t0 A5.svg200px]][6][[File:up2 1 32 t0 D7.svg200px]][12/2][[File:up2 1 32 t0 D6.svg200px]][10]
[[File:up2 1 32 t0 D5.svg200px]][8][[File:up2 1 32 t0 D4.svg200px]][6][[File:up2 1 32 t0 D3.svg200px]][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 end of the 2-length branch leaves the 6-demicube, 131,

Removing the node on the end of the 3-length branch leaves the 122,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 032,

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

E7k-facefkf0f1f2colspan=2f3colspan=3f4colspan=3f5colspan=2f6k-figuresNotesf0f1f2f3f4f5f6
A6( )57635210140210
A3A2A1{ }210080121218
A2A2A1{3}334032023
A3A2{3,3}46420160*
A3A1A1464*302400
A4A2{3,3,3}5101050
D4A1{3,3,4}8243288
A4A1{3,3,3}5101005
D5A1h{4,3,3,3}16801608040
D5168016040800
A5A1{3,3,3,3,3} 61520015
E6{3,32,2} 72720216010801080
D6h{4,3,3,3,3} 32240640160480

The 132 is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.

Rectified 132 polytope

Rectified 132
Type
Schläfli symbol
Coxeter symbol
Coxeter-Dynkin diagram
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter group
Properties

The rectified 132 (also called 0321) is a rectification of the 132 polytope, creating new vertices on the center of edge of the 132. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}.

Alternate names

  • Rectified pentacontahexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (Acronym: lanq) (Jonathan Bowers)

Construction

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, , and the ring represents the position of the active mirror(s).

Removing the node on the end of the 3-length branch leaves the rectified 122 polytope,

Removing the node on the end of the 2-length branch leaves the demihexeract, 131,

Removing the node on the end of the 1-length branch leaves the birectified 6-simplex,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×{},

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

E7k-facefkf0f1colspan=3f2colspan=5f3colspan=6f4colspan=5f5colspan=3f6k-figuresNotesf0f1f2f3f4f5f6
A3A2A1( )1008024241236
A2A1A1{ }2120960213
A2A2013380640**
A2A2A133*40320*0
A2A1A133**1209600
A3A20246400
011612440*
A3A1612404*
A3A1A1612044*
A3A10246004
A4A2021103020100
A4A11030200105
D4A101112496323232
A4021103010020
A4A11030010200
0351000100
D5A1021180480320160160
A5022209060060
D5021180480160160320
A5031156020060
A5A11560020600
E602217206480432021604320
A6032352101400210
D6031124019206406401920

Images

E7E6 / F4B7 / A6A5D7 / B6D6 / B5D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3
[[File:up2 1 32 t1 E7.svg200px]][18][[File:up2 1 32 t1 E6.svg200px]][12][[File:up2 1 32 t1 A6.svg200px]][14]
[[File:up2 1 32 t1 A5.svg200px]][6][[File:up2 1 32 t1 D7.svg200px]][12/2][[File:up2 1 32 t1 D6.svg200px]][10]
[[File:up2 1 32 t1 D5.svg200px]][8][[File:up2 1 32 t1 D4.svg200px]][6][[File:up2 1 32 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]
  • o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - lanq

References

  1. [http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6 and E7 Lattices] {{Webarchive. link. (2016-01-30 , Edward Pervin)
  2. Elte, 1912
  3. Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, p. 202–203
Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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