1 32 polytope

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


321
231
132

Rectified 321
birectified 321

Rectified 231
Rectified 132
Orthogonal projections in E7 Coxeter plane

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
132
TypeUniform 7-polytope
Family1k2 polytope
Schläfli symbol{3,33,2}
Coxeter symbol132
Coxeter diagram
6-faces182:
56 122
126 131
5-faces4284:
756 121
1512 121
2016 {34}
4-faces23688:
4032 {33}
7560 111
12096 {33}
Cells50400:
20160 {32}
30240 {32}
Faces40320 {3}
Edges10080
Vertices576
Vertex figuret2{35}
Petrie polygonOctadecagon
Coxeter groupE7, [33,2,1], order 2903040
Propertiesconvex

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
  • 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
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

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]

E7k-facefkf0f1f2f3f4f5f6k-figuresnotes
A6( ) f0 5763521014021035105105214221772r{3,3,3,3,3}E7/A6 = 72*8!/7! = 576
A3A2A1{ } f1 21008012121841212612343{3,3}x{3}E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A2A1{3} f2 33403202316336132{ }∨{3}E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A3A2{3,3} f3 46420160*13033031{3}∨( )E7/A3A2 = 72*8!/4!/3! = 20160
A3A1A1 464*3024002214122Phyllic disphenoidE7/A3A1A1 = 72*8!/4!/2/2 = 30240
A4A2{3,3,3} f4 51010504032**30030{3}E7/A4A2 = 72*8!/5!/3! = 4032
D4A1{3,3,4} 8243288*7560*12021{ }∨( )E7/D4A1 = 72*8!/8/4!/2 = 7560
A4A1{3,3,3} 5101005**1209602112E7/A4A1 = 72*8!/5!/2 = 12096
D5A1h{4,3,3,3} f5 1680160804016100756**20{ }E7/D5A1 = 72*8!/16/5!/2 = 756
D5 1680160408001016*1512*11E7/D5 = 72*8!/16/5! = 1512
A5A1{3,3,3,3,3} 61520015006**201602E7/A5A1 = 72*8!/6!/2 = 2016
E6{3,32,2} f6 727202160108010802162702162727056*( )E7/E6 = 72*8!/72/6! = 56
D6h{4,3,3,3,3} 3224064016048006019201232*126E7/D6 = 72*8!/32/6! = 126

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] [[3<sup>3,3,1</sup>]] [34,3,1]
Order 48 720 23,040 2,903,040 ∞
Graph - -
Name 13,-1 130 131 132 133 134

Rectified 1_32 polytope
Rectified 132
TypeUniform 7-polytope
Schläfli symbolt1{3,33,2}
Coxeter symbol0321
Coxeter-Dynkin diagram
6-faces758
5-faces12348
4-faces72072
Cells191520
Faces241920
Edges120960
Vertices10080
Vertex figure{3,3}×{3}×{}
Coxeter groupE7, [33,2,1], order 2903040
Propertiesconvex

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

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.[6]

E7k-facefkf0f1f2f3f4f5f6k-figuresnotes
A3A2A1( ) f0 10080242412368123618244121824126681263423{3,3}x{3}x{ }E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A1A1{ } f1 21209602131263313663133621312( )v{3}v{ }E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A2A201 f2 3380640**1130013330033310311{3}v( )v( )E7/A2A2 = 72*8!/3!/3! = 80640
A2A2A1 33*40320*0203010603030601302{3}v{ }E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A2A1A1 33**1209600021201242112421212{ }v{ }v( )E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A3A202 f3 4640020160****13000033000310{3}v( )E7/A3A2 = 72*8!/4!/3! = 20160
011 612440*20160***10300030300301
A3A1 612404**60480**01120012210211SphenoidE7/A3A1 = 72*8!/4!/2 = 60480
A3A1A1 612044***30240*00202010401202{ }v{ }E7/A3A1A1 = 72*8!/4!/2/2 = 30240
A3A102 46004****6048000021101221112SphenoidE7/A3A1 = 72*8!/4!/2 = 60480
A4A2021 f4 103020100550004032*****30000300{3}E7/A4A2 = 72*8!/5!/3! = 4032
A4A1 10302001050500*12096****12000210{ }v()E7/A4A1 = 72*8!/5!/2 = 12096
D4A10111 249632323208880**7560***10200201E7/D4A1 = 72*8!/8/4!/2 = 7560
A4021 10301002000505***24192**01110111( )v( )v( )E7/A4 = 72*8!/5! = 34192
A4A1 10300102000055****12096*00201102{ }v()E7/A4A1 = 72*8!/5!/2 = 12096
03 510001000005*****1209600021012
D5A10211 f5 80480320160160808080400161610000756****200{ }E7/D5A1 = 72*8!/16/5!/2 = 756
A5022 20906006015030015060600*4032***110E7/A5 = 72*8!/6! = 4032
D50211 80480160160320040808080001016160**1512**101E7/D5 = 72*8!/16/5! = 1512
A5031 1560200600015030000606***4032*011E7/A5 = 72*8!/6! = 4032
A5A1 1560020600001530000066****2016002E7/A5A1 = 72*8!/6!/2 = 2016
E60221 f6 72064804320216043201080108021601080108021643227043221602772270056**( )E7/E6 = 72*8!/72/6! = 56
A6032 352101400210350105010502104202107070*576*E7/A6 = 72*8!/7! = 576
D60311 240192064064019200160480480960006019219219200123232**126E7/D6 = 72*8!/32/6! = 126

Images
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
  • List of E7 polytopes

Notes
  1. The Voronoi Cells of the E6* and E7* Lattices, 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
  • 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
    • (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 4-polytope 5-cell 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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