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

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

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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