Wikipedia

1 32 polytope


321
231
132

Rectified 321
birectified 321

Rectified 231
Rectified 132
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: .

1_32 polytope edit

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 names edit

  • 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]

Images edit

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]

Construction edit

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.[4]

E7             k-face fk f0 f1 f2 f3 f4 f5 f6 k-figures notes
A6             ( ) f0 576 35 210 140 210 35 105 105 21 42 21 7 7 2r{3,3,3,3,3} E7/A6 = 72*8!/7! = 576
A3A2A1             { } f1 2 10080 12 12 18 4 12 12 6 12 3 4 3 {3,3}x{3} E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A2A1             {3} f2 3 3 40320 2 3 1 6 3 3 6 1 3 2 { }∨{3} E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A3A2             {3,3} f3 4 6 4 20160 * 1 3 0 3 3 0 3 1 {3}∨( ) E7/A3A2 = 72*8!/4!/3! = 20160
A3A1A1             4 6 4 * 30240 0 2 2 1 4 1 2 2 Phyllic disphenoid E7/A3A1A1 = 72*8!/4!/2/2 = 30240
A4A2             {3,3,3} f4 5 10 10 5 0 4032 * * 3 0 0 3 0 {3} E7/A4A2 = 72*8!/5!/3! = 4032
D4A1             {3,3,4} 8 24 32 8 8 * 7560 * 1 2 0 2 1 { }∨( ) E7/D4A1 = 72*8!/8/4!/2 = 7560
A4A1             {3,3,3} 5 10 10 0 5 * * 12096 0 2 1 1 2 E7/A4A1 = 72*8!/5!/2 = 12096
D5A1             h{4,3,3,3} f5 16 80 160 80 40 16 10 0 756 * * 2 0 { } E7/D5A1 = 72*8!/16/5!/2 = 756
D5             16 80 160 40 80 0 10 16 * 1512 * 1 1 E7/D5 = 72*8!/16/5! = 1512
A5A1             {3,3,3,3,3} 6 15 20 0 15 0 0 6 * * 2016 0 2 E7/A5A1 = 72*8!/6!/2 = 2016
E6             {3,32,2} f6 72 720 2160 1080 1080 216 270 216 27 27 0 56 * ( ) E7/E6 = 72*8!/72/6! = 56
D6             h{4,3,3,3,3} 32 240 640 160 480 0 60 192 0 12 32 * 126 E7/D6 = 72*8!/32/6! = 126

Related polytopes and honeycombs edit

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
1k2 figures in n dimensions
Space Finite Euclidean Hyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group 		E3=A2A1 E4=A4 E5=D5 E6 E7 E8 E9 =   = E8+ E10 =   = E8++
Coxeter
diagram 		                                                                                         
Symmetry
(order) 		[3−1,2,1] [30,2,1] [31,2,1] [[32,2,1]] [33,2,1] [34,2,1] [35,2,1] [36,2,1]
Order 12 120 1,920 103,680 2,903,040 696,729,600 ∞
Graph             - -
Name 1−1,2 102 112 122 132 142 152 162

Rectified 1_32 polytope edit

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 names edit

  • Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym rolin) (Jonathan Bowers)[5]

Construction edit

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.[6]

E7             k-face fk f0 f1 f2 f3 f4 f5 f6 k-figures notes
A3A2A1             ( ) f0 10080 24 24 12 36 8 12 36 18 24 4 12 18 24 12 6 6 8 12 6 3 4 2 3 {3,3}x{3}x{ } E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A1A1             { } f1 2 120960 2 1 3 1 2 6 3 3 1 3 6 6 3 1 3 3 6 2 1 3 1 2 ( )v{3}v{ } E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A2A2             01 f2 3 3 80640 * * 1 1 3 0 0 1 3 3 3 0 0 3 3 3 1 0 3 1 1 {3}v( )v( ) E7/A2A2 = 72*8!/3!/3! = 80640
A2A2A1             3 3 * 40320 * 0 2 0 3 0 1 0 6 0 3 0 3 0 6 0 1 3 0 2 {3}v{ } E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A2A1A1             3 3 * * 120960 0 0 2 1 2 0 1 2 4 2 1 1 2 4 2 1 2 1 2 { }v{ }v( ) E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A3A2             02 f3 4 6 4 0 0 20160 * * * * 1 3 0 0 0 0 3 3 0 0 0 3 1 0 {3}v( ) E7/A3A2 = 72*8!/4!/3! = 20160
            011 6 12 4 4 0 * 20160 * * * 1 0 3 0 0 0 3 0 3 0 0 3 0 1
A3A1             6 12 4 0 4 * * 60480 * * 0 1 1 2 0 0 1 2 2 1 0 2 1 1 Sphenoid E7/A3A1 = 72*8!/4!/2 = 60480
A3A1A1             6 12 0 4 4 * * * 30240 * 0 0 2 0 2 0 1 0 4 0 1 2 0 2 { }v{ } E7/A3A1A1 = 72*8!/4!/2/2 = 30240
A3A1             02 4 6 0 0 4 * * * * 60480 0 0 0 2 1 1 0 1 2 2 1 1 1 2 Sphenoid E7/A3A1 = 72*8!/4!/2 = 60480
A4A2             021 f4 10 30 20 10 0 5 5 0 0 0 4032 * * * * * 3 0 0 0 0 3 0 0 {3} E7/A4A2 = 72*8!/5!/3! = 4032
A4A1             10 30 20 0 10 5 0 5 0 0 * 12096 * * * * 1 2 0 0 0 2 1 0 { }v() E7/A4A1 = 72*8!/5!/2 = 12096
D4A1             0111 24 96 32 32 32 0 8 8 8 0 * * 7560 * * * 1 0 2 0 0 2 0 1 E7/D4A1 = 72*8!/8/4!/2 = 7560
A4             021 10 30 10 0 20 0 0 5 0 5 * * * 24192 * * 0 1 1 1 0 1 1 1 ( )v( )v( ) E7/A4 = 72*8!/5! = 34192
A4A1             10 30 0 10 20 0 0 0 5 5 * * * * 12096 * 0 0 2 0 1 1 0 2 { }v() E7/A4A1 = 72*8!/5!/2 = 12096
            03 5 10 0 0 10 0 0 0 0 5 * * * * * 12096 0 0 0 2 1 0 1 2
D5A1             0211 f5 80 480 320 160 160 80 80 80 40 0 16 16 10 0 0 0 756 * * * * 2 0 0 { } E7/D5A1 = 72*8!/16/5!/2 = 756
A5             022 20 90 60 0 60 15 0 30 0 15 0 6 0 6 0 0 * 4032 * * * 1 1 0 E7/A5 = 72*8!/6! = 4032
D5             0211 80 480 160 160 320 0 40 80 80 80 0 0 10 16 16 0 * * 1512 * * 1 0 1 E7/D5 = 72*8!/16/5! = 1512
A5             031 15 60 20 0 60 0 0 15 0 30 0 0 0 6 0 6 * * * 4032 * 0 1 1 E7/A5 = 72*8!/6! = 4032
A5A1             15 60 0 20 60 0 0 0 15 30 0 0 0 0 6 6 * * * * 2016 0 0 2 E7/A5A1 = 72*8!/6!/2 = 2016
E6             0221 f6 720 6480 4320 2160 4320 1080 1080 2160 1080 1080 216 432 270 432 216 0 27 72 27 0 0 56 * * ( ) E7/E6 = 72*8!/72/6! = 56
A6             032 35 210 140 0 210 35 0 105 0 105 0 21 0 42 0 21 0 7 0 7 0 * 576 * E7/A6 = 72*8!/7! = 576
D6             0311 240 1920 640 640 1920 0 160 480 480 960 0 0 60 192 192 192 0 0 12 32 32 * * 126 E7/D6 = 72*8!/32/6! = 126

Images edit

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 also edit

Notes edit

  1. ^ The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine, Edward Pervin
  2. ^ Elte, 1912
  3. ^ Klitzing, (o3o3o3x *c3o3o3o - lin)
  4. ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
  5. ^ Klitzing, (o3o3x3o *c3o3o3o - rolin)
  6. ^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203

References edit

  • 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
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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