2 21 polytope

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

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 {3^4}
4-faces	648: 432 {3^3} 216 {3^3}
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	fk	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				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}		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 ${\displaystyle \mathbb {C} ^{2}}$ 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
En	3	4	5	6	7	8	9	10
Coxeter group	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8^+	E10 = ${\displaystyle {\bar {T}}_{8}}$ = 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.

2k1 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 = ${\displaystyle {\tilde {E}}_{8}}$ = E8^+	E10 = ${\displaystyle {\bar {T}}_{8}}$ = 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	${\displaystyle {\tilde {E}}_{6}}$=E6^+	E6^++
Coxeter diagram
Graph				∞	∞
Name	22,-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{3^4} 27 t1{3^3,4} 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]	B6 [12/2]
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]	B6 [12/2]
A5 [6]	A4 [5]	A3 / D3 [4]

See also

• List of E6 polytopes

Notes

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)

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	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds