2 21 polytope

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

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.

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 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

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]	[[3^1,2,1]]	[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	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.

Alternate names

Rectified icosihepta-heptacontidi-peton as a rectified 27-72 facetted polypeton (acronym rojak) (Jonathan Bowers)^[6]

Construction

Its construction is based on the E₆ group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

Removing the ring on the short branch leaves the rectified 5-simplex, .

Removing the ring on the end of the other 2-length branch leaves the rectified 5-orthoplex in its alternated form: t₁(2₁₁), .

Removing the ring on the end of the same 2-length branch leaves the 5-demicube: (1₂₁), .

The vertex figure is determined by removing the ringed ring and ringing the neighboring ring. This makes rectified 5-cell prism, t₁{3,3,3}x{}, .

Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow.

Coxeter plane orthographic projections
E6 [12]	D5 [8]	D4 / A2 [6]

A5 [6]	A4 [5]	A3 / D3 [4]

Truncated 2_21 polytope

Truncated 2₂₁ polytope
Type	Uniform 6-polytope
Schläfli symbol	t{3,3,3^2,1}
Coxeter symbol	t(2₂₁)
Coxeter-Dynkin diagram	or
5-faces	72+27+27
4-faces	432+216+432+270
Cells	1080+2160+1080
Faces	720+4320
Edges	216+2160
Vertices	432
Vertex figure	( ) v r{3,3,3}
Coxeter group	E₆, [3^2,2,1], order 51840
Properties	convex

The truncated 2₂₁ has 432 vertices, 5040 edges, 4320 faces, 1350 cells, and 126 4-faces. Its vertex figure is a rectified 5-cell pyramid.

Images

Vertices are colored by their multiplicity in this projection, in progressive order: red, orange, yellow, green, cyan, blue, purple.

Coxeter plane orthographic projections
E6 [12]	D5 [8]	D4 / A2 [6]

A5 [6]	A4 [5]	A3 / D3 [4]

Notes

^ Gosset, 1900
^ Coxeter, H.S.M. (1940). "The Polytope 2₂₁ Whose Twenty-Seven Vertices Correspond to the Lines on the General Cubic Surface". Amer. J. Math. 62 (1): 457–486. doi:10.2307/2371466. JSTOR 2371466.
^ Elte, 1912
^ Klitzing, (x3o3o3o3o *c3o - jak)
^ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
^ Klitzing, (o3x3o3o3o *c3o - rojak)

References

T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
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 [1]
- (Paper 17) Coxeter, The Evolution of Coxeter-Dynkin diagrams, [Nieuw Archief voor Wiskunde 9 (1991) 233-248] See figure 1: (p. 232) (Node-edge graph of polytope)
Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

[1] Gosset, 1900

[2] Coxeter, H.S.M. (1940). "The Polytope 2₂₁ Whose Twenty-Seven Vertices Correspond to the Lines on the General Cubic Surface". Amer. J. Math. 62 (1): 457–486. doi:10.2307/2371466. JSTOR 2371466.

[3] Elte, 1912

[4] Klitzing, (x3o3o3o3o *c3o - jak)

[5] Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203

[6] Klitzing, (o3x3o3o3o *c3o - rojak)

2 21 polytope

2_21 polytope

Alternate names

Coordinates

Construction

Images

Geometric folding

Related complex polyhedra

Related polytopes

Rectified 2_21 polytope

Alternate names

Construction

Images

Truncated 2_21 polytope

Images

See also

Notes

References