# 1 32 polytope

Last updated
 Orthogonal projections in E7 Coxeter plane 321 231 132 Rectified 321 birectified 321 Rectified 231 Rectified 132

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

## Contents

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
Type Uniform 7-polytope
Family 1k2 polytope
Schläfli symbol {3,33,2}
Coxeter symbol 132
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 figure t2{35}
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

• 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
E7E6 / F4B7 / A6

[18]

[12]

[7x2]
A5D7 / B6D6 / B5

[6]

[12/2]

[10]
D5 / B4 / A4D4 / B3 / A2 / G2D3 / 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-facefkf0f1f2f3f4f5f6 k-figures notes
A6( )f0576352101402103510510521422177 2r{3,3,3,3,3} E7/A6 = 72*8!/7! = 576
A3A2A1{ }f121008012121841212612343{3,3}x{3}E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A2A1 {3} f233403202316336132{ }∨{3}E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A3A2 {3,3} f346420160*13033031 {3}∨( ) E7/A3A2 = 72*8!/4!/3! = 20160
A3A1A1464*3024002214122 Phyllic disphenoid E7/A3A1A1 = 72*8!/4!/2/2 = 30240
A4A2 {3,3,3} f451010504032**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
D5A1 h{4,3,3,3} f51680160804016100756**20{ }E7/D5A1 = 72*8!/16/5!/2 = 756
D51680160408001016*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} f6727202160108010802162702162727056*( )E7/E6 = 72*8!/72/6! = 56
D6 h{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
SpaceFiniteEuclideanHyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1A5D6 E7 ${\displaystyle {\tilde {E}}_{7}}$=E7+${\displaystyle {\bar {T}}_{8}}$=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 4872023,0402,903,040
Graph --
Name 13,-1 130 131 132 133 134
1k2 figures in n dimensions
SpaceFiniteEuclideanHyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1E4=A4E5=D5 E6 E7 E8 E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter
diagram
Symmetry
(order)
[3−1,2,1][30,2,1][31,2,1][[3<sup>2,2,1</sup>]][33,2,1][34,2,1][35,2,1][36,2,1]
Order 121201,920103,6802,903,040696,729,600
Graph --
Name 1−1,2 102 112 122 132 142 152 162

## Rectified 1_32 polytope

Rectified 132
Type Uniform 7-polytope
Schläfli symbol t1{3,33,2}
Coxeter symbol 0321
Coxeter-Dynkin diagram
6-faces758
5-faces12348
4-faces72072
Cells191520
Faces241920
Edges120960
Vertices10080
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

• 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-facefkf0f1f2f3f4f5f6 k-figures notes
A3A2A1( )f010080242412368123618244121824126681263423{3,3}x{3}x{ }E7/A3A2A1 = 72*8!/4!/3!/2 = 10080
A2A1A1{ }f121209602131263313663133621312( )v{3}v{ }E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A2A2 01 f23380640**1130013330033310311{3}v( )v( )E7/A2A2 = 72*8!/3!/3! = 80640
A2A2A133*40320*0203010603030601302{3}v{ }E7/A2A2A1 = 72*8!/3!/3!/2 = 40320
A2A1A133**1209600021201242112421212{ }v{ }v( )E7/A2A1A1 = 72*8!/3!/2/2 = 120960
A3A2 02 f34640020160****13000033000310 {3}v( ) E7/A3A2 = 72*8!/4!/3! = 20160
011 612440*20160***10300030300301
A3A1612404**60480**01120012210211 Sphenoid E7/A3A1 = 72*8!/4!/2 = 60480
A3A1A1612044***30240*00202010401202 { }v{ } E7/A3A1A1 = 72*8!/4!/2/2 = 30240
A3A10246004****6048000021101221112SphenoidE7/A3A1 = 72*8!/4!/2 = 60480
A4A2 021 f4103020100550004032*****30000300 {3} E7/A4A2 = 72*8!/5!/3! = 4032
A4A110302001050500*12096****12000210 { }v() E7/A4A1 = 72*8!/5!/2 = 12096
D4A1 0111 249632323208880**7560***10200201E7/D4A1 = 72*8!/8/4!/2 = 7560
A402110301002000505***24192**01110111 ( )v( )v( ) E7/A4 = 72*8!/5! = 34192
A4A110300102000055****12096*00201102 { }v() E7/A4A1 = 72*8!/5!/2 = 12096
03510001000005*****1209600021012
D5A1 0211 f580480320160160808080400161610000756****200{ }E7/D5A1 = 72*8!/16/5!/2 = 756
A5 022 20906006015030015060600*4032***110E7/A5 = 72*8!/6! = 4032
D5021180480160160320040808080001016160**1512**101E7/D5 = 72*8!/16/5! = 1512
A50311560200600015030000606***4032*011E7/A5 = 72*8!/6! = 4032
A5A11560020600001530000066****2016002E7/A5A1 = 72*8!/6!/2 = 2016
E6 0221 f672064804320216043201080108021601080108021643227043221602772270056**( )E7/E6 = 72*8!/72/6! = 56
A6 032 352101400210350105010502104202107070*576*E7/A6 = 72*8!/7! = 576
D6 0311 240192064064019200160480480960006019219219200123232**126E7/D6 = 72*8!/32/6! = 126

### Images

Coxeter plane projections
E7E6 / F4B7 / A6

[18]

[12]

[14]
A5D7 / B6D6 / B5

[6]

[12/2]

[10]
D5 / B4 / A4D4 / B3 / A2 / G2D3 / B2 / A3

[8]

[6]

[4]

## Notes

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

## Related Research Articles

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In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams.

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V72 (for its 72 vertices).

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

In 8-dimensional geometry, the 251 honeycomb is a space-filling uniform tessellation. It is composed of 241 polytope and 8-simplex facets arranged in an 8-demicube vertex figure. It is the final figure in the 2k1 family.

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

In 7-dimensional geometry, the 331 honeycomb is a uniform honeycomb, also given by Schläfli symbol {3,3,3,33,1} and is composed of 321 and 7-simplex facets, with 56 and 576 of them respectively around each vertex.

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets.

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

In geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. The symbol 521 is from Coxeter, named for the length of the 3 branches of its Coxeter-Dynkin diagram.

In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E10) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded.

In geometry, the simplectic honeycomb is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is given a Schläfli symbol {3[n+1]}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of n+1 nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an n-simplex honeycomb is an expanded n-simplex.

## 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
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds