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).^[1]

Coxeter named it 1₂₂ for 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: .

Contents 1 1_22 polytope 1.1 Alternate names 1.2 Construction 1.3 Images 1.4 Related polytopes 1.4.1 Geometric folding 1.4.2 Tessellations 2 Rectified 1_22 polytope 2.1 Alternate names 2.2 Construction 2.3 Images 3 Birectified 1_22 polytope 3.1 Alternate names 3.2 Images 4 See also 5 Notes 6 References

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 hexateron: 022
Petrie polygon	Dodecagon
Coxeter group	E6, [[32,2,1]]
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)^[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, .

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]	A3 / D3 [4]
(2,3,6)	(1,2)	(1,6,8,12)

Related polytopes

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.

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	t2{3,3,32,1} t1{3,32,2}
Coxeter symbol	t1(122)
Coxeter-Dynkin diagram
5-faces	54
4-faces	702
Cells	2160
Faces	2160
Edges	720
Vertices	72
Vertex figure	3-3 duoprism prism
Petrie polygon	Dodecagon
Coxeter group	E6, [[32,2,1]]
Properties	convex

Alternate names

• Birectified 2₂₁ polytope
• Rectified pentacontatetrapeton (Acronym Ram) - rectified 54-facetted polypeton (Jonathan Bowers)^[3]

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}x{3}x{}, .

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

Rectified 122 polytope
Type	Uniform 6-polytope
Schläfli symbol	t2{3,32,2}
Coxeter symbol	t2(122)
Coxeter-Dynkin diagram
5-faces
4-faces
Cells
Faces
Edges	12960
Vertices	2160
Vertex figure
Coxeter group	E6, [[32,2,1]]
Properties	convex

Alternate names

• Bicantellated 2₂₁
• Small rhombated pentacontitetrapeton (Jonathan Bowers)^[4]

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]

See also

• List of E6 polytopes

Notes

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, 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, o3x3o3x3o4o - siborg

Family	An	BCn	Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square		Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	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
n-polytopes	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes