1 32 polytope

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 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 figuret2{35}
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

• E. L. 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]

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,

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]

Related polytopes and honeycombs

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 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 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)[4]

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}×{},

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]

Notes

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)

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 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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