Английская Википедия:2 21 polytope

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Файл:Up 2 21 t0 E6.svg 221 Шаблон:CDD	Файл:Up 2 21 t1 E6.svg Rectified 221 Шаблон:CDD
Файл:Up 1 22 t0 E6.svg (122) Шаблон:CDD	Файл:Up 2 21 t2 E6.svg Birectified 221 (Rectified 122) Шаблон:CDD
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. 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 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: Шаблон:CDD.

2_21 polytope

221 polytope
Type	Uniform 6-polytope
Family	k21 polytope
Schläfli symbol	{3,3,32,1}
Coxeter symbol	221
Coxeter-Dynkin diagram	Шаблон:CDD or Шаблон:CDD
5-faces	99 total: 27 211Файл:5-orthoplex.svg 72 {34}Файл:5-simplex t0.svg
4-faces	648: 432 {33}Файл:4-simplex t0.svg 216 {33}Файл:4-simplex t0.svg
Cells	1080 {3,3}Файл:3-simplex t0.svg
Faces	720 {3}Файл:2-simplex t0.svg
Edges	216
Vertices	27
Vertex figure	121 (5-demicube)
Petrie polygon	Dodecagon
Coxeter group	E6, [32,2,1], 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.
• Icosihepta-heptacontidi-peton - 27-72 facetted polypeton (acronym jak) (Jonathan Bowers)

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, Шаблон:CDD.

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

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

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), Шаблон:CDD. The edge-figure is the vertex figure of the vertex figure, a rectified 5-cell, (0₂₁ polytope), Шаблон:CDD.

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

E6	Шаблон:CDD	k-face	fk	f0	f1	f2	f3	f4		f5		k-figure	notes
D5	Шаблон:CDD	( )	f0	27	16	80	160	80	40	16	10	h{4,3,3,3}	E6/D5 = 51840/1920 = 27
A4A1	Шаблон:CDD	{ }	f1	2	216	10	30	20	10	5	5	r{3,3,3}	E6/A4A1 = 51840/120/2 = 216
A2A2A1	Шаблон:CDD	{3}	f2	3	3	720	6	6	3	2	3	{3}x{ }	E6/A2A2A1 = 51840/6/6/2 = 720
A3A1	Шаблон:CDD	{3,3}	f3	4	6	4	1080	2	1	1	2	{ }v( )	E6/A3A1 = 51840/24/2 = 1080
A4	Шаблон:CDD	{3,3,3}	f4	5	10	10	5	432	*	1	1	{ }	E6/A4 = 51840/120 = 432
A4A1	Шаблон:CDD			5	10	10	5	*	216	0	2		E6/A4A1 = 51840/120/2 = 216
A5	Шаблон:CDD	{3,3,3,3}	f5	6	15	20	15	6	0	72	*	( )	E6/A5 = 51840/720 = 72
D5	Шаблон:CDD	{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]
Файл:Up 2 21 t0 E6.svg (1,3)	Файл:Up 2 21 t0 D5.svg (1,3)	Файл:Up 2 21 t0 D4.svg (3,9)	Файл:Up 2 21 t0 B6.svg (1,3)
A5 [6]	A4 [5]	A3 / D3 [4]
Файл:Up 2 21 t0 A5.svg (1,3)	Файл:Up 2 21 t0 A4.svg (1,2)	Файл:Up 2 21 t0 D3.svg (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 Шаблон:Dynkin	F4 Шаблон:Dynkin2
Файл:E6 graph.svg 221 Шаблон:CDD	Файл:24-cell t3 F4.svg 24-cell Шаблон:CDD

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

Related complex polyhedra

The regular complex polygon ₃{3}₃{3}₃, Шаблон:CDD, in <math>\mathbb{C}^2</math> has a real representation as the 2₂₁ polytope, Шаблон:CDD, 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. Шаблон:K 21 polytopes

The 2₂₁ polytope is fourth in dimensional series 2_k2. Шаблон:2 k1 polytopes

The 2₂₁ polytope is second in dimensional series 2_2k. Шаблон:2 2k polytopes

Rectified 2_21 polytope

Rectified 221 polytope
Type	Uniform 6-polytope
Schläfli symbol	t1{3,3,32,1}
Coxeter symbol	t1(221)
Coxeter-Dynkin diagram	Шаблон:CDD or Шаблон:CDD
5-faces	126 total: 72 t1{34} Файл:5-simplex t1.svg 27 t1{33,4} Файл:5-cube t3.svg 27 t1{3,32,1} Файл:5-demicube t0 D5.svg
4-faces	1350
Cells	4320
Faces	5040
Edges	2160
Vertices	216
Vertex figure	rectified 5-cell prism
Coxeter group	E6, [32,2,1], 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)

Construction

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

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

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

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

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{}, Шаблон:CDD.

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]
Файл:Up 2 21 t1 E6.svg	Файл:Up 2 21 t1 D5.svg	Файл:Up 2 21 t1 D4.svg	Файл:Up 2 21 t1 B6.svg
A5 [6]	A4 [5]	A3 / D3 [4]
Файл:Up 2 21 t1 A5.svg	Файл:Up 2 21 t1 A4.svg	Файл:Up 2 21 t1 D3.svg

Truncated 2_21 polytope

Truncated 221 polytope
Type	Uniform 6-polytope
Schläfli symbol	t{3,3,32,1}
Coxeter symbol	t(221)
Coxeter-Dynkin diagram	Шаблон:CDD or Шаблон:CDD
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, [32,2,1], 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]
Файл:Up 2 21 t01 E6.svg	Файл:Up 2 21 t01 D5.svg	Файл:Up 2 21 t01 D4.svg	Файл:Up 2 21 t01 B6.svg
A5 [6]	A4 [5]	A3 / D3 [4]
Файл:Up 2 21 t01 A5.svg	Файл:Up 2 21 t01 A4.svg	Файл:Up 2 21 t01 D3.svg

See also

• List of E6 polytopes

References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• Шаблон:Citation
• 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, Шаблон:ISBN [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)
• Шаблон:KlitzingPolytopes x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak