1₂₂ polytope

orthogonal projections in E₆ Coxeter plane
1₂₂	Rectified 1₂₂	Birectified 1₂₂
2₂₁	Rectified 2₂₁	Birectified 1₂₂

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	1_k2 polytope
Schläfli symbol	{3,3^2,2}
Coxeter symbol	1₂₂
Coxeter-Dynkin diagram	or
5-faces	54: 27 1₂₁ 27 1₂₁
4-faces	702: 270 1₁₁ 432 1₂₀
Cells	2160: 1080 1₁₀ 1080 {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-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]

Images

Coxeter plane orthographic 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, 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	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	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
A₃A₁	{3,3}	f₃	4	6	4	*	1080	0	1	2	1	2	{ }∨( )	E₆/A₃A₁ = 72*6!/4!/2 = 1080
A₄A₁	{3,3,3}	f₄	5	10	10	5	0	216	*	*	2	0	{ }	E₆/A₄A₁ = 72*6!/5!/2 = 216
A₄A₁	{3,3,3}		5	10	10	0	5	*	216	*	0	2		E₆/A₄A₁ = 72*6!/5!/2 = 216
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
D₅	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

The regular complex polyhedron ₃{3}₃{4}₂, , in $\mathbb {C} ^{2}$ 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: .

1_k2 figures in n dimensions
Space	Finite						Euclidean	Hyperbolic
n	3	4	5	6	7	8	9	10
Coxeter group	E₃=A₂A₁	E₄=A₄	E₅=D₅	E₆	E₇	E₈	E₉ = ${\tilde {E}}_{8}$ = E₈⁺	E₁₀ = ${\bar {T}}_{8}$ = E₈⁺⁺
Coxeter diagram
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]