2₂₁ polytope

In 6-dimensional geometry, the 2₂₁ polytope is a uniform 6-polytope, constructed within the symmetry of the E₆ group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure.[1] It is also called the Schläfli polytope.

orthogonal projections in E₆ Coxeter plane
2₂₁	Rectified 2₂₁
(1₂₂)	Birectified 2₂₁ (Rectified 1₂₂)

Its Coxeter symbol is 2₂₁, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences. He also studied[2] its connection with the 27 lines on the cubic surface, which are naturally in correspondence with the vertices of 2₂₁.

The rectified 2₂₁ is constructed by points at the mid-edges of the 2₂₁. The birectified 2₂₁ is constructed by points at the triangle face centers of the 2₂₁, and is the same as the rectified 1₂₂.

These polytopes are a part of family of 39 convex uniform polytopes in 6-dimensions, made of uniform 5-polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

2_21 polytope

2₂₁ polytope
Type	Uniform 6-polytope
Family	k₂₁ polytope
Schläfli symbol	{3,3,3^2,1}
Coxeter symbol	2₂₁
Coxeter-Dynkin diagram	or
5-faces	99 total: 27 2₁₁ 72 {3⁴}
4-faces	648: 432 {3³} 216 {3³}
Cells	1080 {3,3}
Faces	720 {3}
Edges	216
Vertices	27
Vertex figure	1₂₁ (5-demicube)
Petrie polygon	Dodecagon
Coxeter group	E₆, [3^2,2,1], order 51840
Properties	convex

The 2₂₁ has 27 vertices, and 99 facets: 27 5-orthoplexes and 72 5-simplices. Its vertex figure is a 5-demicube.

For visualization this 6-dimensional polytope is often displayed in a special skewed orthographic projection direction that fits its 27 vertices within a 12-gonal regular polygon (called a Petrie polygon). Its 216 edges are drawn between 2 rings of 12 vertices, and 3 vertices projected into the center. Higher elements (faces, cells, etc.) can also be extracted and drawn on this projection.

The Schläfli graph is the 1-skeleton of this polytope.

Alternate names

E. L. Elte named it V₂₇ (for its 27 vertices) in his 1912 listing of semiregular polytopes.[3]
Icosihepta-heptacontidi-peton - 27-72 facetted polypeton (acronym jak) (Jonathan Bowers)[4]

Coordinates

The 27 vertices can be expressed in 8-space as an edge-figure of the 4₂₁ polytope:

(-2,0,0,0,-2,0,0,0), (0,-2,0,0,-2,0,0,0), (0,0,-2,0,-2,0,0,0), (0,0,0,-2,-2,0,0,0), (0,0,0,0,-2,0,0,-2), (0,0,0,0,0,-2,-2,0)
(2,0,0,0,-2,0,0,0), (0,2,0,0,-2,0,0,0), (0,0,2,0,-2,0,0,0), (0,0,0,2,-2,0,0,0), (0,0,0,0,-2,0,0,2)
(-1,-1,-1,-1,-1,-1,-1,-1), (-1,-1,-1, 1,-1,-1,-1, 1), (-1,-1, 1,-1,-1,-1,-1, 1), (-1,-1, 1, 1,-1,-1,-1,-1), (-1, 1,-1,-1,-1,-1,-1, 1), (-1, 1,-1, 1,-1,-1,-1,-1), (-1, 1, 1,-1,-1,-1,-1,-1), (1,-1,-1,-1,-1,-1,-1, 1) (1,-1, 1,-1,-1,-1,-1,-1), (1,-1,-1, 1,-1,-1,-1,-1) (1, 1,-1,-1,-1,-1,-1,-1), (-1, 1, 1, 1,-1,-1,-1, 1) (1,-1, 1, 1,-1,-1,-1, 1) (1, 1,-1, 1,-1,-1,-1, 1) (1, 1, 1,-1,-1,-1,-1, 1) (1, 1, 1, 1,-1,-1,-1,-1)

Construction

Its construction is based on the E₆ group.

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

Removing the node on the short branch leaves the 5-simplex, .

Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form: (2₁₁), .

Every simplex facet touches a 5-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 5-demicube (1₂₁ polytope), . The edge-figure is the vertex figure of the vertex figure, a rectified 5-cell, (0₂₁ polytope), .

Seen in a configuration matrix, the element counts can be derived from the Coxeter group orders.[5]

E₆	k-face	f_k	f₀	f₁	f₂	f₃	f₄		f₅		k-figure	notes
D₅	( )	f₀	27	16	80	160	80	40	16	10	h{4,3,3,3}	E₆/D₅ = 51840/1920 = 27
A₄A₁	{ }	f₁	2	216	10	30	20	10	5	5	r{3,3,3}	E₆/A₄A₁ = 51840/120/2 = 216
A₂A₂A₁	{3}	f₂	3	3	720	6	6	3	2	3	{3}x{ }	E₆/A₂A₂A₁ = 51840/6/6/2 = 720
A₃A₁	{3,3}	f₃	4	6	4	1080	2	1	1	2	{ }v( )	E₆/A₃A₁ = 51840/24/2 = 1080
A₄	{3,3,3}	f₄	5	10	10	5	432	*	1	1	{ }	E₆/A₄ = 51840/120 = 432
A₄A₁	{3,3,3}	f₄	5	10	10	5	*	216	0	2	{ }	E₆/A₄A₁ = 51840/120/2 = 216
A₅	{3,3,3,3}	f₅	6	15	20	15	6	0	72	*	( )	E₆/A₅ = 51840/720 = 72
D₅	{3,3,3,4}	f₅	10	40	80	80	16	16	*	27	( )	E₆/D₅ = 51840/1920 = 27

Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow. The number of vertices by color are given in parentheses.

Coxeter plane orthographic projections
E6 [12]	D5 [8]	D4 / A2 [6]	B6 [12/2]
(1,3)	(1,3)	(3,9)	(1,3)
A5 [6]	A4 [5]	A3 / D3 [4]
(1,3)	(1,2)	(1,4,7)

Geometric folding

The 2₂₁ is related to the 24-cell by a geometric folding of the E6/F4 Coxeter-Dynkin diagrams. This can be seen in the Coxeter plane projections. The 24 vertices of the 24-cell are projected in the same two rings as seen in the 2₂₁.

E₆	F₄
2₂₁	24-cell

This polytope can tessellate Euclidean 6-space, forming the 2₂₂ honeycomb with this Coxeter-Dynkin diagram: .

Related complex polyhedra

The regular complex polygon ₃{3}₃{3}₃, , in $\mathbb {C} ^{2}$ has a real representation as the 2₂₁ polytope, , in 4-dimensional space. It is called a Hessian polyhedron after Edmund Hess. It has 27 vertices, 72 3-edges, and 27 3{3}3 faces. Its complex reflection group is ₃[3]₃[3]₃, order 648.

Related polytopes

The 2₂₁ is fourth 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 dimensional
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

The 2₂₁ polytope is fourth in dimensional series 2_k2.

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<sup>1,2,1</sup>]]	[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	2₃₁	2₄₁	2₅₁	2₆₁

The 2₂₁ polytope is second in dimensional series 2_2k.

2_2k figures of n dimensions
Space	Finite			Euclidean	Hyperbolic
n	4	5	6	7	8
Coxeter group	A₂A₂	A₅	E₆	${\tilde {E}}_{6}$ =E₆⁺	E₆⁺⁺
Coxeter diagram
Graph				∞	∞
Name	2_2,-1	2₂₀	2₂₁	2₂₂	2₂₃

Rectified 2_21 polytope

Rectified 2₂₁ polytope
Type	Uniform 6-polytope
Schläfli symbol	t₁{3,3,3^2,1}
Coxeter symbol	t₁(2₂₁)
Coxeter-Dynkin diagram	or
5-faces	126 total: 72 t₁{3⁴} 27 t₁{3³,4} 27 t₁{3,3^2,1}
4-faces	1350
Cells	4320
Faces	5040
Edges	2160
Vertices	216
Vertex figure	rectified 5-cell prism
Coxeter group	E₆, [3^2,2,1], order 51840
Properties	convex

The rectified 2₂₁ has 216 vertices, and 126 facets: 72 rectified 5-simplices, and 27 rectified 5-orthoplexes and 27 5-demicubes . Its vertex figure is a rectified 5-cell prism.