# 1 32 polytope

 Orthogonal projections in E6 Coxeter plane 321 Rectified 321 Birectified 321 Rectified 132 132 231 Rectified 231

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
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}
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

• 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 ${\tilde{E}}_{7}$=E7+ ${\bar{T}}_8$=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 = ${\tilde{E}}_{8}$ = E8+ E10 = ${\bar{T}}_8$ = 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 12 120 192 103,680 2,903,040 696,729,600
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-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

• 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 [1]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Richard Klitzing, 7D, uniform polytopes (polyexa) o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin