1₂₂ polytope

Template:Short description

orthogonal projections in E₆ Coxeter plane
File:Up 1 22 t0 E6.svg 1₂₂ Template:CDD	File:Up 1 22 t1 E6.svg Rectified 1₂₂ Template:CDD	File:Up 1 22 t2 E6.svg Birectified 1₂₂ Template:CDD
Script error: No such module "Unsubst". Trirectified 1₂₂ Template:CDD	File:Up 1 22 t01 E6.svg Truncated 1₂₂ Template:CDD
File:Up 2 21 t0 E6.svg 2₂₁ Template:CDD	File:Up 2 21 t1 E6.svg Rectified 2₂₁ Template:CDD

In 6-dimensional geometry, the 1₂₂ polytope is a uniform polytope, constructed from the E₆ group. It was first published in E. L. Elte's 1912 listing of semiregular polytopes, named as V₇₂ (for its 72 vertices).^[1]

Its Coxeter symbol is 1₂₂, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 1₂₂, constructed by positions points on the elements of 1₂₂. The rectified 1₂₂ is constructed by points at the mid-edges of the 1₂₂. The birectified 1₂₂ is constructed by points at the triangle face centers of the 1₂₂.

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

1₂₂ polytope

1₂₂ polytope
Type	Uniform 6-polytope
Family	1_k2 polytope
Schläfli symbol	{3,3^2,2}
Coxeter symbol	1₂₂
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
5-faces	54: 27 1₂₁ File:Demipenteract graph ortho.svg 27 1₂₁ File:Demipenteract graph ortho.svg
4-faces	702: 270 1₁₁ File:Cross graph 4.svg 432 1₂₀ File:4-simplex t0.svg
Cells	2160: 1080 1₁₀ File:3-simplex t0.svg 1080 {3,3} File:3-simplex t0.svg
Faces	2160 {3} File:2-simplex t0.svg
Edges	720
Vertices	72
Vertex figure	Birectified 5-simplex: 0₂₂ File:5-simplex t2.svg
Petrie polygon	Dodecagon
Coxeter group	E₆, [[3,3^2,2]], order 103680
Properties	convex, isotopic

The 1₂₂ polytope contains 72 vertices, and 54 5-demicubic facets. It has a birectified 5-simplex vertex figure. Its 72 vertices represent the root vectors of the simple Lie group E₆.

Alternate names

Pentacontatetrapeton (Acronym: mo) - 54-facetted polypeton (Jonathan Bowers)^[2]

Images

Coxeter plane orthographic projections
E6 [12]	D5 [8]	D4 / A2 [6]
File:Up 1 22 t0 E6.svg (1,2)	File:Up 1 22 t0 D5.svg (1,3)	File:Up 1 22 t0 D4.svg (1,9,12)
B6 [12/2]	A5 [6]	A4 [[5]] = [10]	A3 / D3 [4]
File:Up 1 22 t0 B6.svg (1,2)	File:Up 1 22 t0 A5.svg (2,3,6)	File:Up 1 22 t0 A4.svg (1,2)	File:Up 1 22 t0 D3.svg (1,6,8,12)

Construction

It is created by a Wythoff construction upon a set of 6 hyperplane mirrors in 6-dimensional space.

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

Removing the node on either of 2-length branches leaves the 5-demicube, 1₃₁, Template:CDD.

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 0₂₂, Template:CDD.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^[3]

E₆	Template:CDD	k-face	f_k	f₀	f₁	f₂	f₃		f₄			f₅		k-figure	notes
A₅	Template:CDD	( )	f₀	72	20	90	60	60	15	15	30	6	6	r{3,3,3}	E₆/A₅ = 72*6!/6! = 72
A₂A₂A₁	Template:CDD	{ }	f₁	2	720	9	9	9	3	3	9	3	3	{3}×{3}	E₆/A₂A₂A₁ = 72*6!/3!/3!/2 = 720
A₂A₁A₁	Template:CDD	{3}	f₂	3	3	2160	2	2	1	1	4	2	2	s{2,4}	E₆/A₂A₁A₁ = 72*6!/3!/2/2 = 2160
A₃A₁	Template:CDD	{3,3}	f₃	4	6	4	1080	*	1	0	2	2	1	{ }∨( )	E₆/A₃A₁ = 72*6!/4!/2 = 1080
A₃A₁	Template:CDD	{3,3}	f₃	4	6	4	*	1080	0	1	2	1	2	{ }∨( )	E₆/A₃A₁ = 72*6!/4!/2 = 1080
A₄A₁	Template:CDD	{3,3,3}	f₄	5	10	10	5	0	216	*	*	2	0	{ }	E₆/A₄A₁ = 72*6!/5!/2 = 216
A₄A₁	Template:CDD	{3,3,3}		5	10	10	0	5	*	216	*	0	2		E₆/A₄A₁ = 72*6!/5!/2 = 216
D₄	Template:CDD	h{4,3,3}		8	24	32	8	8	*	*	270	1	1		E₆/D₄ = 72*6!/8/4! = 270
D₅	Template:CDD	h{4,3,3,3}	f₅	16	80	160	80	40	16	0	10	27	*	( )	E₆/D₅ = 72*6!/16/5! = 27
D₅	Template:CDD	h{4,3,3,3}	f₅	16	80	160	40	80	0	16	10	*	27	( )	E₆/D₅ = 72*6!/16/5! = 27

Related complex polyhedron

File:Complex polyhedron 3-3-3-4-2.png

Orthographic projection in Aut(E6) Coxeter plane with 18-gonal symmetry for complex polyhedron, ₃{3}₃{4}₂. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces.

The regular complex polyhedron ₃{3}₃{4}₂, Template:CDD, in $ℂ^{2}$ has a real representation as the 1₂₂ polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is ₃[3]₃[4]₂, order 1296. It has a half-symmetry quasiregular construction as Template:CDD, as a rectification of the Hessian polyhedron, Template:CDD.^[4]

Related polytopes and honeycomb

Along with the semiregular polytope, 2₂₁, it is also one of a family of 39 convex uniform polytopes in 6-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: Template:CDD.

Template:1 k2 polytopes

Geometric folding

The 1₂₂ is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to 1₂₂ in 6 dimensions, F4 to the 24-cell in 4 dimensions. 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 1₂₂.

E6/F4 Coxeter planes
File:Up 1 22 t0 E6.svg 1₂₂	File:24-cell t3 F4.svg 24-cell
D4/B4 Coxeter planes
File:Up 1 22 t0 D4.svg 1₂₂	File:24-cell t3 B3.svg 24-cell

Tessellations

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, 2₂₂, Template:CDD.

Rectified 1₂₂ polytope

Rectified 1₂₂
Type	Uniform 6-polytope
Schläfli symbol	2r{3,3,3^2,1} r{3,3^2,2}
Coxeter symbol	0₂₂₁
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
5-faces	126
4-faces	1566
Cells	6480
Faces	6480
Edges	6480
Vertices	720
Vertex figure	3-3 duoprism prism
Petrie polygon	Dodecagon
Coxeter group	E₆, [[3,3^2,2]], order 103680
Properties	convex

The rectified 1₂₂ polytope (also called 0₂₂₁) can tessellate 6-dimensional space as the Voronoi cell of the E6* honeycomb lattice (dual of E6 lattice).^[5]

Alternate names

Birectified 2₂₁ polytope
Rectified pentacontatetrapeton (Acronym: ram) - rectified 54-facetted polypeton (Jonathan Bowers)^[6]

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 1 22 t1 E6.svg	File:Up 1 22 t1 D5.svg	File:Up 1 22 t1 D4.svg	File:Up 1 22 t1 B6.svg
A5 [6]	A4 [5]	A3 / D3 [4]
File:Up 1 22 t1 A5.svg	File:Up 1 22 t1 A4.svg	File:Up 1 22 t1 D3.svg

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 birectified 5-simplex, Template:CDD.

Removing the ring on either of 2-length branches leaves the birectified 5-orthoplex in its alternated form: t₂(2₁₁), Template:CDD.

The vertex figure is determined by removing the ringed node and ringing the neighboring ring. This makes 3-3 duoprism prism, {3}×{3}×{}, Template:CDD.

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^[3]Template:R

E₆	Template:CDD	k-face	f_k	f₀	f₁	f₂			f₃					f₄					f₅			k-figure	notes
A₂A₂A₁	Template:CDD	( )	f₀	720	18	18	18	9	6	18	9	6	9	6	3	6	9	3	2	3	3	{3}×{3}×{ }	E₆/A₂A₂A₁ = 72*6!/3!/3!/2 = 720
A₁A₁A₁	Template:CDD	{ }	f₁	2	6480	2	2	1	1	4	2	1	2	2	1	2	4	1	1	2	2	{ }∨{ }∨( )	E₆/A₁A₁A₁ = 72*6!/2/2/2 = 6480
A₂A₁	Template:CDD	{3}	f₂	3	3	4320	*	*	1	2	1	0	0	2	1	1	2	0	1	2	1	Sphenoid	E₆/A₂A₁ = 72*6!/3!/2 = 4320
A₂A₁	Template:CDD			3	3	*	4320	*	0	2	0	1	1	1	0	2	2	1	1	1	2	Sphenoid	E₆/A₂A₁ = 72*6!/3!/2 = 4320
A₂A₁A₁	Template:CDD			3	3	*	*	2160	0	0	2	0	2	0	1	0	4	1	0	2	2	{ }∨{ }	E₆/A₂A₁A₁ = 72*6!/3!/2/2 = 2160
A₂A₁	Template:CDD	{3,3}	f₃	4	6	4	0	0	1080	*	*	*	*	2	1	0	0	0	1	2	0	{ }∨( )	E₆/A₂A₁ = 72*6!/3!/2 = 1080
A₃	Template:CDD	r{3,3}		6	12	4	4	0	*	2160	*	*	*	1	0	1	1	0	1	1	1	{3}	E₆/A₃ = 72*6!/4! = 2160
A₃A₁	Template:CDD	r{3,3}		6	12	4	0	4	*	*	1080	*	*	0	1	0	2	0	0	2	1	{ }∨( )	E₆/A₃A₁ = 72*6!/4!/2 = 1080
	Template:CDD	{3,3}		4	6	0	4	0	*	*	*	1080	*	0	0	2	0	1	1	0	2
	Template:CDD	r{3,3}		6	12	0	4	4	*	*	*	*	1080	0	0	0	2	1	0	1	2
A₄	Template:CDD	r{3,3,3}	f₄	10	30	20	10	0	5	5	0	0	0	432	*	*	*	*	1	1	0	{ }	E₆/A₄ = 72*6!/5! = 432
A₄A₁	Template:CDD			10	30	20	0	10	5	0	5	0	0	*	216	*	*	*	0	2	0		E₆/A₄A₁ = 72*6!/5!/2 = 216
A₄	Template:CDD			10	30	10	20	0	0	5	0	5	0	*	*	432	*	*	1	0	1		E₆/A₄ = 72*6!/5! = 432
D₄	Template:CDD	{3,4,3}		24	96	32	32	32	0	8	8	0	8	*	*	*	270	*	0	1	1		E₆/D₄ = 72*6!/8/4! = 270
A₄A₁	Template:CDD	r{3,3,3}		10	30	0	20	10	0	0	0	5	5	*	*	*	*	216	0	0	2		E₆/A₄A₁ = 72*6!/5!/2 = 216
A₅	Template:CDD	2r{3,3,3,3}	f₅	20	90	60	60	0	15	30	0	15	0	6	0	6	0	0	72	*	*	( )	E₆/A₅ = 72*6!/6! = 72
D₅	Template:CDD	2r{4,3,3,3}		80	480	320	160	160	80	80	80	0	40	16	16	0	10	0	*	27	*		E₆/D₅ = 72*6!/16/5! = 27
D₅	Template:CDD	2r{4,3,3,3}		80	480	160	320	160	0	80	40	80	80	0	0	16	10	16	*	*	27		E₆/D₅ = 72*6!/16/5! = 27

Truncated 1₂₂ polytope

Truncated 1₂₂
Type	Uniform 6-polytope
Schläfli symbol	t{3,3^2,2}
Coxeter symbol	t(1₂₂)
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
5-faces	72+27+27
4-faces	32+216+432+270+216
Cells	1080+2160+1080+1080+1080
Faces	4320+4320+2160
Edges	6480+720
Vertices	1440
Vertex figure	( )v{3}x{3}
Petrie polygon	Dodecagon
Coxeter group	E₆, [[3,3^2,2]], order 103680
Properties	convex

Alternate names

Truncated 1₂₂ polytope (Acronym: tim)^[7]

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.

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 1 22 t01 E6.svg	File:Up 1 22 t01 D5.svg	File:Up 1 22 t01 D4.svg	File:Up 1 22 t01 B6.svg
A5 [6]	A4 [5]	A3 / D3 [4]
File:Up 1 22 t01 A5.svg	File:Up 1 22 t01 A4.svg	File:Up 1 22 t01 D3.svg

Birectified 1₂₂ polytope

Birectified 1₂₂ polytope
Type	Uniform 6-polytope
Schläfli symbol	2r{3,3^2,2}
Coxeter symbol	2r(1₂₂)
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
5-faces	126
4-faces	2286
Cells	10800
Faces	19440
Edges	12960
Vertices	2160
Vertex figure
Coxeter group	E₆, [[3,3^2,2]], order 103680
Properties	convex

Alternate names

Bicantellated 2₂₁
Birectified pentacontatetrapeton (barm) (Jonathan Bowers)^[8]

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 1 22 t2 E6.svg	File:Up 1 22 t2 D5.svg	File:Up 1 22 t2 D4.svg	File:Up 1 22 t2 B6.svg
A5 [6]	A4 [5]	A3 / D3 [4]
File:Up 1 22 t2 A5.svg	File:Up 1 22 t2 A4.svg	File:Up 1 22 t2 D3.svg

Trirectified 1₂₂ polytope

Trirectified 1₂₂ polytope
Type	Uniform 6-polytope
Schläfli symbol	3r{3,3^2,2}
Coxeter symbol	3r(1₂₂)
Coxeter-Dynkin diagram	Template:CDD or Template:CDD
5-faces	558
4-faces	4608
Cells	8640
Faces	6480
Edges	2160
Vertices	270
Vertex figure
Coxeter group	E₆, [[3,3^2,2]], order 103680
Properties	convex

Alternate names

Tricantellated 2₂₁
Trirectified pentacontatetrapeton (Acronym: trim, old: cacam, tram, mak) (Jonathan Bowers)^[9]

Notes

↑ Elte, 1912
↑ Klitzing, (o3o3o3o3o *c3x - mo)
↑ ^a ^b Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, pp. 202–203
↑ Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47
↑ The Voronoi Cells of the E6* and E7* Lattices Template:Webarchive, Edward Pervin
↑ Klitzing, (o3o3x3o3o *c3o - ram)
↑ Klitzing, (o3o3x3o3o *c3x - tim)
↑ Klitzing, (o3x3o3x3o *c3o - barm)
↑ Klitzing, (x3o3o3o3x *c3o - trim)

Script error: No such module "Check for unknown parameters".

References

Script error: No such module "citation/CS1".
H. S. M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45] See p334 (figure 3.6a) by Peter mcMullen: (12-gonal node-edge graph of 1₂₂)
Template:KlitzingPolytopes o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3o3x3o3o *c3x - tim, o3x3o3x3o *c3o - barm, x3o3o3o3x *c3o - trim

Template:Polytopes

[1] Elte, 1912

[2] Klitzing, (o3o3o3o3o *c3x - mo)

[:0-3] Coxeter, Regular Polytopes, 11.8 Gosset figures in six, seven, and eight dimensions, pp. 202–203

[4] Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47

[5] The Voronoi Cells of the E6* and E7* Lattices Template:Webarchive, Edward Pervin

[ram-6] Klitzing, (o3o3x3o3o *c3o - ram)

[7] Klitzing, (o3o3x3o3o *c3x - tim)

[8] Klitzing, (o3x3o3x3o *c3o - barm)

[9] Klitzing, (x3o3o3o3x *c3o - trim)

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1₂₂ polytope

Contents

1₂₂ polytope

Alternate names

Images

Construction

Related complex polyhedron

Related polytopes and honeycomb

Geometric folding

Tessellations

Rectified 1₂₂ polytope

Alternate names

Images

Construction

Truncated 1₂₂ polytope

Alternate names

Construction

Images

Birectified 1₂₂ polytope

Alternate names

Images

Trirectified 1₂₂ polytope

Alternate names

See also

Notes

References

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1 22 polytope

122 polytope

Alternate names

Images

Construction

Related complex polyhedron

Related polytopes and honeycomb

Geometric folding

Tessellations

Rectified 122 polytope

Alternate names

Images

Construction

Truncated 122 polytope

Alternate names

Construction

Images

Birectified 122 polytope

Alternate names

Images

Trirectified 122 polytope

Alternate names

See also

Notes

References

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1₂₂ polytope

1₂₂ polytope

Rectified 1₂₂ polytope

Truncated 1₂₂ polytope

Birectified 1₂₂ polytope

Trirectified 1₂₂ polytope