1 32 polytope

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.

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	1_k2 polytope
Schläfli symbol	{3,3^3,2}
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, [3^3,2,1], 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]

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	${\tilde {E}}_{7}$ =E₇⁺	${\bar {T}}_{8}$ =E₇⁺⁺
Coxeter diagram
Symmetry	[3^−1,3,1]	[3^0,3,1]	[3^1,3,1]	[3^2,3,1]	[[3<sup>3,3,1</sup>]]	[3^4,3,1]
Order	48	720	23,040	2,903,040	∞
Graph					-	-
Name	1_3,-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 = ${\tilde {E}}_{8}$ = E₈⁺	E₁₀ = ${\bar {T}}_{8}$ = E₈⁺⁺
Coxeter diagram
Symmetry (order)	[3^−1,2,1]	[3^0,2,1]	[3^1,2,1]	[[3<sup>2,2,1</sup>]]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
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{3,3^3,2}
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, [3^3,2,1], 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}×{}.