1 32 polytope

321		231		132
Rectified 321			birectified 321
Rectified 231			Rectified 132
Orthogonal projections in E6 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
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}
Petrie polygon	Octadecagon
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 1_3k series. The next figure is the Euclidean honeycomb 1₃₃ and the final is a noncompact hyperbolic honeycomb, 1₃₄.

1_3k 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

1_k2 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 = = E8+	E10 = = 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	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]

See also

• List of E7 polytopes

Notes

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]
• Richard Klitzing, 7D, uniform polytopes (polyexa) o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin