2₃₁ polytope

Orthogonal projections in E₇ Coxeter plane
3₂₁	2₃₁		1₃₂
Rectified 3₂₁		Birectified 3₂₁
Rectified 2₃₁		Rectified 1₃₂

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

Its Coxeter symbol is 2₃₁, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.

The rectified 2₃₁ is constructed by points at the mid-edges of the 2₃₁.

These polytopes are part of a family of 127 (or 2⁷−1) convex uniform polytopes in seven dimensions, made of uniform polytope facets and vertex figures, defined by all combinations of rings in this Coxeter-Dynkin diagram: .

2₃₁ polytope

Gosset 2₃₁ polytope
Type	Uniform 7-polytope
Family	2_k1 polytope
Schläfli symbol	{3,3,3^3,1}
Coxeter symbol	2₃₁
Coxeter diagram
6-faces	632: 56 2₂₁ 576 {3⁵}
5-faces	4788: 756 2₁₁ 4032 {3⁴}
4-faces	16128: 4032 2₀₁ 12096 {3³}
Cells	20160 {3²}
Faces	10080 {3}
Edges	2016
Vertices	126
Vertex figure	1₃₁
Petrie polygon	Octadecagon
Coxeter group	E₇, [3^3,2,1]
Properties	convex

The 2₃₁ is composed of 126 vertices, 2016 edges, 10080 faces (triangles), 20160 cells (tetrahedra), 16128 4-faces (4-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 2₂₁). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E₇.

This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, 3₃₁.

Alternate names

E. L. Elte named it V₁₂₆ (for its 126 vertices) in his 1912 listing of semiregular polytopes.^[1]
It was called 2₃₁ by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
Pentacontahexa-pentacosiheptacontahexa-exon (Acronym: laq) - 56-576 facetted polyexon (Jonathan Bowers)^[2]

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 short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 3₂₁ polytope, .

Removing the node on the end of the 3-length branch leaves the 2₂₁. There are 56 of these facets. These facets are centered on the locations of the vertices of the 1₃₂ polytope, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 1₃₁, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^[3]

E₇	k-face	f_k	f₀	f₁	f₂	f₃	f₄		f₅		f₆		k-figures	Notes
D₆	( )	f₀	126	32	240	640	160	480	60	192	12	32	6-demicube	E₇/D₆ = 72x8!/32/6! = 126
A₅A₁	{ }	f₁	2	2016	15	60	20	60	15	30	6	6	rectified 5-simplex	E₇/A₅A₁ = 72x8!/6!/2 = 2016
A₃A₂A₁	{3}	f₂	3	3	10080	8	4	12	6	8	4	2	tetrahedral prism	E₇/A₃A₂A₁ = 72x8!/4!/3!/2 = 10080
A₃A₂	{3,3}	f₃	4	6	4	20160	1	3	3	3	3	1	tetrahedron	E₇/A₃A₂ = 72x8!/4!/3! = 20160
A₄A₂	{3,3,3}	f₄	5	10	10	5	4032	*	3	0	3	0	{3}	E₇/A₄A₂ = 72x8!/5!/3! = 4032
A₄A₁	{3,3,3}	f₄	5	10	10	5	*	12096	1	2	2	1	Isosceles triangle	E₇/A₄A₁ = 72x8!/5!/2 = 12096
D₅A₁	{3,3,3,4}	f₅	10	40	80	80	16	16	756	*	2	0	{ }	E₇/D₅A₁ = 72x8!/32/5! = 756
A₅	{3,3,3,3}	f₅	6	15	20	15	0	6	*	4032	1	1	{ }	E₇/A₅ = 72x8!/6! = 7287 = 4032
E₆	{3,3,3^2,1}	f₆	27	216	720	1080	216	432	27	72	56	*	( )	E₇/E₆ = 72x8!/72x6! = 8*7 = 56
A₆	{3,3,3,3,3}	f₆	7	21	35	35	0	21	0	7	*	576	( )	E₇/A₆ = 72x8!/7! = 72×8 = 576

Images

Coxeter plane projections
E7	E6 / F4	B6 / 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

2_k1 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	[3^−1,2,1]	[3^0,2,1]	[[3^1,2,1]]	[3^2,2,1]	[3^3,2,1]	[3^4,2,1]	[3^5,2,1]	[3^6,2,1]
Order	12	120	384	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	2_−1,1	2₀₁	2₁₁	2₂₁		2₄₁	2₅₁	2₆₁