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1 32 polytope

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

Contents

1_32 polytopeEdit

132
Type Uniform 7-polytope
Family 1k2 polytope
Schläfli symbol {3,33,2}
Coxeter symbol 132
Coxeter diagram            
6-faces 182:
56 122 
126 131 
5-faces 4284:
756 121 
1512 121 
2016 {34} 
4-faces 23688:
4032 {33} 
7560 111 
12096 {33} 
Cells 50400:
20160 {32} 
30240 {32} 
Faces 40320 {3} 
Edges 10080
Vertices 576
Vertex figure t2{35}  
Petrie polygon Octadecagon
Coxeter group E7, [33,2,1], order 2903040
Properties convex

This polytope can tessellate 7-dimensional space, with symbol 133, and Coxeter-Dynkin diagram,              . It is the Voronoi cell of the dual E7* lattice.[1]

Alternate namesEdit

  • Emanuel Lodewijk Elte named it V576 (for its 576 vertices) in his 1912 listing of semiregular polytopes.[2]
  • Coxeter called it 132 for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
  • Pentacontihexa-hecatonicosihexa-exon (Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers)[3]

ConstructionEdit

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,            

ImagesEdit

Coxeter plane projections
E7 E6 / F4 B7 / 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]

Related polytopes and honeycombsEdit

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.

13k dimensional figures
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1 A5 D6 E7  =E7+  =E7++
Coxeter
diagram
                                                                   
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [[33,3,1]] [34,3,1]
Order 48 720 23,040 2,903,040
Graph       - -
Name 13,-1 130 131 132 133 134

Rectified 1_32 polytopeEdit

Rectified 132
Type Uniform 7-polytope
Schläfli symbol t1{3,33,2}
Coxeter symbol 0321
Coxeter-Dynkin diagram            
6-faces 758
5-faces 12348
4-faces 72072
Cells 191520
Faces 241920
Edges 120960
Vertices 10080
Vertex figure {3,3}×{3}×{}
Coxeter group E7, [33,2,1], order 2903040
Properties convex

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 namesEdit

  • Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym rolin) (Jonathan Bowers)[4]

ConstructionEdit

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}×{},            

ImagesEdit

Coxeter plane projections
E7 E6 / F4 B7 / A6
 
[18]
 
[12]
 
[14]
A5 D7 / B6 D6 / B5
 
[6]
 
[12/2]
 
[10]
D5 / B4 / A4 D4 / B3 / A2 / G2 D3 / B2 / A3
 
[8]
 
[6]
 
[4]

See alsoEdit

NotesEdit

  1. ^ The Voronoi Cells of the E6* and E7* Lattices, Edward Pervin
  2. ^ Elte, 1912
  3. ^ Klitzing, (o3o3o3x *c3o3o3o - lin)
  4. ^ Klitzing, (o3o3x3o *c3o3o3o - rolin)

ReferencesEdit

  • 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)".  o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin
Fundamental convex regular and uniform polytopes in dimensions 2–10
An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Triangle Square p-gon Hexagon Pentagon
Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
5-simplex 5-orthoplex5-cube 5-demicube
6-simplex 6-orthoplex6-cube 6-demicube 122221
7-simplex 7-orthoplex7-cube 7-demicube 132231321
8-simplex 8-orthoplex8-cube 8-demicube 142241421
9-simplex 9-orthoplex9-cube 9-demicube
10-simplex 10-orthoplex10-cube 10-demicube
n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds