1 22 polytope

122	Rectified 122	Birectified 122
221	Rectified 221
orthogonal projections in E6 Coxeter 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).

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, construcated 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: .

1_22 polytope

122 polytope
Type	Uniform 6-polytope
Family	1k2 polytope
Schläfli symbol	{3,32,2}
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, [[3,32,2]], order 103680
Properties	convex, isotopic

The 1_22 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)

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, .

Images

Coxeter plane orthographic projections

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

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
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]	[30,2,1]	[31,2,1]	[[32,2,1]]	[33,2,1]	[34,2,1]	[35,2,1]	[36,2,1]
Order	12	120	192	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 1_22 polytope

Rectified 122
Type	Uniform 6-polytope
Schläfli symbol	2r{3,3,32,1} r{3,32,2}
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, [[3,32,2]], 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).

Alternate names

• Birectified 2₂₁ polytope
• Rectified pentacontatetrapeton (acronym Ram) - rectified 54-facetted polypeton (Jonathan Bowers)

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, {3}×{3}×{}, .

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]
A5 [6]	A4 [5]	A3 / D3 [4]

Birectified 1_22 polytope

Birectified 122 polytope
Type	Uniform 6-polytope
Schläfli symbol	2r{3,32,2}
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, [[3,32,2]], order 103680
Properties	convex

Alternate names

• Bicantellated 2₂₁
• Birectified pentacontitetrapeton (barm) (Jonathan Bowers)

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

Trirectified 1_22 polytope

Trirectified 122 polytope
Type	Uniform 6-polytope
Schläfli symbol	3r{3,32,2}
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, [[3,32,2]], order 103680
Properties	convex

Alternate names

• Tricantellated 2₂₁
• Trirectified pentacontitetrapeton (trim) (Jonathan Bowers)

See also

• List of E6 polytopes

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, ISBN 978-0-471-01003-6 [1]
  • (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)
• Richard Klitzing, 6D, uniform polytopes (polypeta) o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3x3o3x3o *c3o - barm