2₂₁ polytope

Orthogonal projections in E₆ Coxeter plane
File:Up 2 21 t0 E6.svg 2₂₁ Template:CDD	File:Up 2 21 t1 E6.svg Rectified 2₂₁ Template:CDD	File:Up 2 21 t01 E6.svg Truncated 2₂₁ Template:CDD
File:Up 1 22 t0 E6.svg (1₂₂) Template:CDD	File:Up 2 21 t2 E6.svg Birectified 2₂₁ (Rectified 1₂₂) Template:CDD

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: Template:CDD.

2₂₁ polytope

2₂₁ polytope
Type	Uniform 6-polytope
Family	k₂₁ polytope
Schläfli symbol	{3,3,3^2,1}
Coxeter symbol	2₂₁
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
5-faces	99 total: 27 2₁₁ File:5-orthoplex.svg 72 {3⁴} File:5-simplex t0.svg
4-faces	648: 432 {3³} File:4-simplex t0.svg 216 {3³} File:4-simplex t0.svg
Cells	1080 {3,3} File:3-simplex t0.svg
Faces	720 {3}File:2-simplex t0.svg
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-heptacontadi-peton - 27-72 facetted polypeton (Acronym: jak) (Jonathan Bowers)Template:Sfn

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, Template:CDD. Removing the node on the short branch leaves the 5-simplex, Template:CDD. Removing the node on the end of the 2-length branch leaves the 5-orthoplex in its alternated form: (2₁₁), Template: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), Template:CDD. The edge-figure is the vertex figure of the vertex figure, a rectified 5-cell, (0₂₁ polytope), Template:CDD.

Seen in a configuration matrix, the element counts can be derived from the Coxeter group orders.^[4]

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

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

Related complex polyhedra

The regular complex polygon ₃{3}₃{3}₃, Template:CDD, in $ℂ^{2}$ has a real representation as the 2₂₁ polytope, Template: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. Template:K 21 polytopes

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

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

Rectified 2₂₁ polytope

Rectified 2₂₁ polytope
Type	Uniform 6-polytope
Schläfli symbol	t₁{3,3,3^2,1}
Coxeter symbol	t₁(2₂₁)
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
5-faces	126 total: 72 t₁{3⁴} File:5-simplex t1.svg 27 t₁{3³,4} File:5-cube t3.svg 27 t₁{3,3^2,1} File: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, [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-heptacontadi-peton as a rectified 27-72 facetted polypeton (Acronym: rojak) (Jonathan Bowers)Template:Sfn

Construction

Its construction is based on the E₆ group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: Template:CDD. Removing the ring on the short branch leaves the rectified 5-simplex, Template:CDD. Removing the ring on the end of the other 2-length branch leaves the rectified 5-orthoplex in its alternated form: t₁(2₁₁), Template:CDD. Removing the ring on the end of the same 2-length branch leaves the 5-demicube: (1₂₁), Template: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{}, Template: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]
File:Up 2 21 t1 E6.svg	File:Up 2 21 t1 D5.svg	File:Up 2 21 t1 D4.svg	File:Up 2 21 t1 B6.svg
A5 [6]	A4 [5]	A3 / D3 [4]
File:Up 2 21 t1 A5.svg	File:Up 2 21 t1 A4.svg	File:Up 2 21 t1 D3.svg

Truncated 2₂₁ polytope

Truncated 2₂₁ polytope
Type	Uniform 6-polytope
Schläfli symbol	t{3,3,3^2,1}
Coxeter symbol	t(2₂₁)
Coxeter-Dynkin diagram	Template:CDD or Template: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, [3^2,2,1], order 51840
Properties	convex

The truncated 2₂₁ has 432 vertices, 2376 edges, 5040 faces, 4320 cells, 1350 4-faces, and 126 5-faces. Its vertex figure is a rectified 5-cell pyramid.

Alternate names

Truncated icosihepta-heptacontadi-peton as a truncated 27-72 facetted polypeton (Acronym: tojak)Template:Sfn

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]
File:Up 2 21 t01 E6.svg	File:Up 2 21 t01 D5.svg	File:Up 2 21 t01 D4.svg	File:Up 2 21 t01 B6.svg
A5 [6]	A4 [5]	A3 / D3 [4]
File:Up 2 21 t01 A5.svg	File:Up 2 21 t01 A4.svg	File:Up 2 21 t01 D3.svg

Notes

↑ Gosset, 1900
↑ Script error: No such module "Citation/CS1".
↑ Elte, 1912
↑ Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, pp. 202–203

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References

T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
Script error: No such module "citation/CS1".
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, wiley.com, Template:Isbn
- (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)
Template:KlitzingPolytopes x3o3o3o3o *c3o - jak, o3x3o3o3o *c3o - rojak, x3x3o3o3o *c3o - tojak Template:Sfn whitelist

Template:Polytopes

[1] Gosset, 1900

[2] Script error: No such module "Citation/CS1".

[3] Elte, 1912

[4] Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, pp. 202–203

[1]

[2]

[3]

[4]

2₂₁ polytope

Contents

2₂₁ polytope

Alternate names

Coordinates

Construction

Images

Geometric folding

Related complex polyhedra

Related polytopes

Rectified 2₂₁ polytope

Alternate names

Construction

Images

Truncated 2₂₁ polytope

Alternate names

Images

See also

Notes

References

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2 21 polytope

221 polytope

Alternate names

Coordinates

Construction

Images

Geometric folding

Related complex polyhedra

Related polytopes

Rectified 221 polytope

Alternate names

Construction

Images

Truncated 221 polytope

Alternate names

Images

See also

Notes

References

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2₂₁ polytope

2₂₁ polytope

Rectified 2₂₁ polytope

Truncated 2₂₁ polytope