1 22 polytope explained

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: .

1₂₂ polytope

bgcolor=#e7dcc3 colspan=2	1₂₂ polytope
Type	Uniform 6-polytope
Family	1_k2 polytope
Schläfli symbol
Coxeter symbol	1₂₂
Coxeter-Dynkin diagram	or
5-faces
4-faces
Cells
Faces
Edges	720
Vertices	72
Vertex figure	Birectified 5-simplex 0₂₂
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

Pentacontatetra-peton (Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)^[2]

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, .

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

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

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

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

Related complex polyhedron

The regular complex polyhedron ₃₃₂,, in

C²

has a real representation as the 1₂₂ polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 33 faces. Its complex reflection group is ₃[3]₃[4]₂, order 1296. It has a half-symmetry quasiregular construction as, as a rectification of the Hessian polyhedron, .^[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: .

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.

Tessellations

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

Rectified 1₂₂ polytope

bgcolor=#e7dcc3 colspan=2	Rectified 1₂₂
Type	Uniform 6-polytope
Schläfli symbol	2r r
Coxeter symbol	0₂₂₁
Coxeter-Dynkin diagram	or
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.

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 birectified 5-simplex, .

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

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

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^[7]

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

Truncated 1₂₂ polytope

bgcolor=#e7dcc3 colspan=2	Truncated 1₂₂
Type	Uniform 6-polytope
Schläfli symbol	t
Coxeter symbol	t(1₂₂)
Coxeter-Dynkin diagram	or
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	vx
Petrie polygon	Dodecagon
Coxeter group	E₆, 3,3^2,2, order 103680
Properties	convex

Alternate names

Truncated 1₂₂ polytope

Construction

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

Images

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

Birectified 1₂₂ polytope

bgcolor=#e7dcc3 colspan=2	Birectified 1₂₂ polytope
Type	Uniform 6-polytope
Schläfli symbol	2r
Coxeter symbol	2r(1₂₂)
Coxeter-Dynkin diagram	or
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 pentacontitetrapeton (barm) (Jonathan Bowers)^[8]

Images

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

Trirectified 1₂₂ polytope

bgcolor=#e7dcc3 colspan=2	Trirectified 1₂₂ polytope
Type	Uniform 6-polytope
Schläfli symbol	3r
Coxeter symbol	3r(1₂₂)
Coxeter-Dynkin diagram	or
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 pentacontitetrapeton (trim or cacam) (Jonathan Bowers)^[9]

References

- 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, http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
- (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)
o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3x3o3x3o *c3o - barm

Notes and References

Elte, 1912
Klitzing, (o3o3o3o3o *c3x - mo)
Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
Coxeter, H. S. M., Regular Complex Polytopes, second edition, Cambridge University Press, (1991). p.30 and p.47
http://home.digital.net/~pervin/publications/vermont.html The Voronoi Cells of the E6* and E7* Lattices
Klitzing, (o3o3x3o3o *c3o - ram)
Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
Klitzing, (o3x3o3x3o *c3o - barm)
Klitzing, (x3o3o3o3x *c3o - cacam

1 22 polytope explained

122 polytope

Alternate names

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

References

Notes and References

1₂₂ polytope

Rectified 1₂₂ polytope

Truncated 1₂₂ polytope

Birectified 1₂₂ polytope

Trirectified 1₂₂ polytope