1 22 polytope

122	Rectified 122	Birectified 122
Trirectified 122	Truncated 122
221	Rectified 221
Orthogonal projections in E6Coxeter plane

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

122 polytope
Type	Uniform 6-polytope
Family	1k2 polytope
Schläfli symbol	{3,3^2,2}
Coxeter symbol	122
Coxeter-Dynkin diagram	or
5-faces	54:27 12127 121
4-faces	702:270 111432 120
Cells	2160:1080 1101080 {3,3}
Faces	2160 {3}
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-simplexvertex figure. Its 72 vertices represent the root vectors of the simple Lie groupE₆.

Alternate names

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

Images

Coxeter planeorthographic projections

E6[12]	D5[8]	D4 / A2[6]
(1,2)	(1,3)	(1,9,12)
B6[12/2]	A5[6]	A4[[5]] = [10]	A3 / D3[4]
(1,2)	(2,3,6)	(1,2)	(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, .

Removing the node on either of 2-length branches leaves the 5-demicube, 1₂₁, .

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

Related complex polyhedron

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 polyhedron₃{3}₃{4}₂, , in ${\displaystyle \mathbb {C} ^{2}}$ 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 , 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: .

1k2 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
n	3	4	5	6	7	8	9	10
Coxetergroup	E3=A2A1	E4=A4	E5=D5	E6	E7	E8	E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8^+	E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8^++
Coxeterdiagram
Symmetry(order)	[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	103,680	2,903,040	696,729,600	∞
Graph							-	-
Name	1−1,2	102	112	122	132	142	152	162

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
122	24-cell
D4/B4 Coxeter planes
122	24-cell

Tessellations

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

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	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.

Coxeter planeorthographic projections

E6[12]	D5[8]	D4 / A2[6]	B6[12/2]
A5[6]	A4[5]	A3 / D3[4]

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 either of 2-length branches 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, {3}×{3}×{}, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^[3][6]

E6		k-face	fk	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	{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		{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
				3	3	*	4320	*	0	2	0	1	1	1	0	2	2	1	1	1	2
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		{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		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				6	12	4	0	4	*	*	1080	*	*	0	1	0	2	0	0	2	1	{ }∨( )	E6/A3A1 = 72*6!/4!/2 = 1080
		{3,3}		4	6	0	4	0	*	*	*	1080	*	0	0	2	0	1	1	0	2
		r{3,3}		6	12	0	4	4	*	*	*	*	1080	0	0	0	2	1	0	1	2
A4		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				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		{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		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		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		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
				80	480	160	320	160	0	80	40	80	80	0	0	16	10	16	*	*	27

Truncated 1₂₂ polytope

Truncated 122
Type	Uniform 6-polytope
Schläfli symbol	t{3,3^2,2}
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	( )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: .

Images

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

Coxeter planeorthographic projections

E6[12]	D5[8]	D4 / A2[6]	B6[12/2]
A5[6]	A4[5]	A3 / D3[4]

Birectified 1₂₂ polytope

Birectified 122 polytope
Type	Uniform 6-polytope
Schläfli symbol	2r{3,3^2,2}
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 pentacontatetrapeton (barm) (Jonathan Bowers)^[8]

Images

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

Coxeter planeorthographic projections

E6[12]	D5[8]	D4 / A2[6]	B6[12/2]
A5[6]	A4[5]	A3 / D3[4]

Trirectified 1₂₂ polytope

Trirectified 122 polytope
Type	Uniform 6-polytope
Schläfli symbol	3r{3,3^2,2}
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 pentacontatetrapeton (Acronym: trim, old: cacam, tram, mak) (Jonathan Bowers)^[9]

See also

• List of E6 polytopes

Notes

1. Elte, 1912
2. Klitzing, (o3o3o3o3o *c3x - mo)
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* LatticesArchived 2016-01-30 at the Wayback Machine, Edward Pervin
6. Klitzing, (o3o3x3o3o *c3o - ram)
7. Klitzing, (o3o3x3o3o *c3x - tim)
8. Klitzing, (o3x3o3x3o *c3o - barm)
9. Klitzing, (x3o3o3o3x *c3o - trim)

References

• Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen
• 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, ISBN 978-0-471-01003-6
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45], p. 334 (figure 3.6a) by Peter mcMullen: (12-gonal node-edge graph of 1₂₂)
• Klitzing, Richard. "6D uniform polytopes (polypeta) with acronyms". o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3o3x3o3o *c3x - tim, o3x3o3x3o *c3o - barm, x3o3o3o3x *c3o - trim

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 • Polytope operations