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

In 7-dimensional geometry, the 3₂₁ polytope is a uniform 7-polytope, constructed within the symmetry of the E₇ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.^[1]

Its Coxeter symbol is 3₂₁, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 3-node sequences.

The rectified 3₂₁ is constructed by points at the mid-edges of the 3₂₁. The birectified 3₂₁ is constructed by points at the triangle face centers of the 3₂₁. The trirectified 3₂₁ is constructed by points at the tetrahedral centers of the 3₂₁, and is the same as the rectified 1₃₂.

These polytopes are part of a family of 127 (2⁷-1) convex uniform polytopes in 7-dimensions, made of uniform 6-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

3₂₁ polytope

3₂₁ polytope
Type	Uniform 7-polytope
Family	k₂₁ polytope
Schläfli symbol	{3,3,3,3^2,1}
Coxeter symbol	3₂₁
Coxeter diagram
6-faces	702 total: 126 3₁₁ 576 {3⁵}
5-faces	6048: 4032 {3⁴} 2016 {3⁴}
4-faces	12096 {3³}
Cells	10080 {3,3}
Faces	4032 {3}
Edges	756
Vertices	56
Vertex figure	2₂₁ polytope
Petrie polygon	octadecagon
Coxeter group	E₇, [3^3,2,1], order 2903040
Properties	convex

In 7-dimensional geometry, the 3₂₁ polytope is a uniform polytope. It has 56 vertices, and 702 facets: 126 3₁₁ and 576 6-simplexes.

For visualization this 7-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 56 vertices within an 18-gonal regular polygon (called a Petrie polygon). Its 756 edges are drawn between 3 rings of 18 vertices, and 2 vertices in the center. Specific higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

The 1-skeleton of the 3₂₁ polytope is the Gosset graph.

This polytope, along with the 7-simplex, can tessellate 7-dimensional space, represented by 3₃₁ and Coxeter-Dynkin diagram: .

Alternate names

It is also called the Hess polytope for Edmund Hess who first discovered it.
It was enumerated by Thorold Gosset in his 1900 paper. He called it an 7-ic semi-regular figure.^[1]
E. L. Elte named it V₅₆ (for its 56 vertices) in his 1912 listing of semiregular polytopes.^[2]
H.S.M. Coxeter called it 3₂₁ due to its bifurcating Coxeter-Dynkin diagram, having 3 branches of length 3, 2, and 1, and having a single ring on the final node of the 3 branch.
Hecatonicosihexa-pentacosiheptacontihexa-exon (Acronym Naq) - 126-576 facetted polyexon (Jonathan Bowers)^[3]

Coordinates

The 56 vertices can be most simply represented in 8-dimensional space, obtained by the 28 permutations of the coordinates and their opposite:

± (-3, -3, 1, 1, 1, 1, 1, 1)

Construction

Its construction is based on the E7 group. Coxeter named it as 3₂₁ by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 3-node sequence.

The facet information can be extracted from its Coxeter-Dynkin diagram, .

Removing the node on the short branch leaves the 6-simplex, .

Removing the node on the end of the 2-length branch leaves the 6-orthoplex in its alternated form: 3₁₁, .

Every simplex facet touches a 6-orthoplex facet, while alternate facets of the orthoplex touch either a simplex or another orthoplex.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes 2₂₁ polytope, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^[4]

E₇	k-face	f_k	f₀	f₁	f₂	f₃	f₄	f₅		f₆		k-figures	notes
E₆	( )	f₀	56	27	216	720	1080	432	216	72	27	2₂₁	E₇/E₆ = 72x8!/72x6! = 56
D₅A₁	{ }	f₁	2	756	16	80	160	80	40	16	10	5-demicube	E₇/D₅A₁ = 72x8!/16/5!/2 = 756
A₄A₂	{3}	f₂	3	3	4032	10	30	20	10	5	5	rectified 5-cell	E₇/A₄A₂ = 72x8!/5!/2 = 4032
A₃A₂A₁	{3,3}	f₃	4	6	4	10080	6	6	3	2	3	triangular prism	E₇/A₃A₂A₁ = 72x8!/4!/3!/2 = 10080
A₄A₁	{3,3,3}	f₄	5	10	10	5	12096	2	1	1	2	isosceles triangle	E₇/A₄A₁ = 72x8!/5!/2 = 12096
A₅A₁	{3,3,3,3}	f₅	6	15	20	15	6	4032	*	1	1	{ }	E₇/A₅A₁ = 72x8!/6!/2 = 4032
A₅	{3,3,3,3}	f₅	6	15	20	15	6	*	2016	0	2	{ }	E₇/A₅ = 72x8!/6! = 2016
A₆	{3,3,3,3,3}	f₆	7	21	35	35	21	10	0	576	*	( )	E₇/A₆ = 72x8!/7! = 576
D₆	{3,3,3,3,4}	f₆	12	60	160	240	192	32	32	*	126	( )	E₇/D₆ = 72x8!/32/6! = 126

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

The 3₂₁ is fifth in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.

k₂₁ figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
E_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	1,920	51,840	2,903,040	696,729,600	∞
Graph							-	-
Name	−1₂₁	0₂₁	1₂₁	2₂₁	3₂₁	4₂₁	5₂₁	6₂₁

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3_k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

3_k1 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]] = [4,3,3,3,3]	[3^2,3,1]	[3^3,3,1]	[3^4,3,1]
Order	48	720	46,080	2,903,040	∞
Graph					-	-
Name	3_1,-1	3₁₀	3₁₁	3₂₁	3₃₁	3₄₁