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1 32 polytope

Uniform polytope

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
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