# 1 22 polytope

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

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]

## Contents

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: .

## 1_22 polytope

122 polytope
Type Uniform 6-polytope
Family 1k2 polytope
Schläfli symbol {3,32,2}
Coxeter symbol122
Coxeter-Dynkin diagram or
5-faces54:
27 121
27 121
4-faces702:
270 111
432 120
Cells2160:
1080 110
1080 {3,3}
Faces2160 {3}
Edges720
Vertices72
Vertex figure Birectified 5-simplex:
022
Petrie polygon Dodecagon
Coxeter group E6, [[3,3<sup>2,2</sup>]], 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) [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]

E6k-facefkf0f1f2f3f4f5k-figurenotes
A5( )f0722090606015153066 r{3,3,3} E6/A5 = 72*6!/6! = 72
A2A2A1{ }f1272099933933 {3}×{3} E6/A2A2A1 = 72*6!/3!/3!/2 = 720
A2A1A1 {3} f23321602211422 s{2,4} E6/A2A1A1 = 72*6!/3!/2/2 = 2160
A3A1 {3,3} f34641080*10221 { }∨( ) E6/A3A1 = 72*6!/4!/2 = 1080
464*108001212
A4A1 {3,3,3} f45101050216**20{ }E6/A4A1 = 72*6!/5!/2 = 216
5101005*216*02
D4 h{4,3,3} 8243288**27011E6/D4 = 72*6!/8/4! = 270
D5 h{4,3,3,3} f5168016080401601027*( )E6/D5 = 72*6!/16/5! = 27
1680160408001610*27

The regular complex polyhedron 3{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]

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
SpaceFiniteEuclideanHyperbolic
n 3 4 5 6 7 8 9 10
Coxeter
group
E3=A2A1E4=A4E5=D5 E6 E7 E8 E9 = ${\displaystyle {\tilde {E}}_{8}}$ = E8+E10 = ${\displaystyle {\bar {T}}_{8}}$ = E8++
Coxeter
diagram
Symmetry
(order)
[3−1,2,1][30,2,1][31,2,1][[3<sup>2,2,1</sup>]][33,2,1][34,2,1][35,2,1][36,2,1]
Order 121201,920103,6802,903,040696,729,600
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-faces126
4-faces1566
Cells6480
Faces6480
Edges6480
Vertices720
Vertex figure 3-3 duoprism prism
Petrie polygon Dodecagon
Coxeter group E6, [[3,3<sup>2,2</sup>]], 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 plane orthographic 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 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}×{}, .

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders. [7] [8]

E6k-facefkf0f1f2f3f4f5k-figurenotes
A2A2A1( )f0720181818961896963693233 {3}×{3}×{ } E6/A2A2A1 = 72*6!/3!/3!/2 = 720
A1A1A1{ }f1264802211421221241122 { }∨{ }∨( ) E6/A1A1A1 = 72*6!/2/2/2 = 6480
A2A1 {3} f2334320**1210021120121 Sphenoid E6/A2A1 = 72*6!/3!/2 = 4320
33*4320*0201110221112
A2A1A133**21600020201041022 { }∨{ } E6/A2A1A1 = 72*6!/3!/2/2 = 2160
A2A1 {3,3} f3464001080****21000120 { }∨( ) E6/A2A1 = 72*6!/3!/2 = 1080
A3 r{3,3} 612440*2160***10110111 {3} E6/A3 = 72*6!/4! = 2160
A3A1612404**1080**01020021 { }∨( ) E6/A3A1 = 72*6!/4!/2 = 1080
{3,3} 46040***1080*00201102
r{3,3} 612044****108000021012
A4 r{3,3,3} f410302010055000432****110{ }E6/A4 = 72*6!/5! = 432
A4A110302001050500*216***020E6/A4A1 = 72*6!/5!/2 = 216
A410301020005050**432**101E6/A4 = 72*6!/5! = 432
D4 h{4,3,3} 249632323208808***270*011E6/D4 = 72*6!/8/4! = 270
A4A1 r{3,3,3} 10300201000055****216002E6/A4A1 = 72*6!/5!/2 = 216
A5 2r{3,3,3,3} f5209060600153001506060072**( )E6/A5 = 72*6!/6! = 72
D5 rh{4,3,3,3} 8048032016016080808004016160100*27*E6/D5 = 72*6!/16/5! = 27
8048016032016008040808000161016**27

## Truncated 1_22 polytope

Truncated 122
Type Uniform 6-polytope
Schläfli symbol t{3,32,2}
Coxeter symbol t(122)
Coxeter-Dynkin diagram
or
5-faces72+27+27
4-faces32+216+432+270+216
Cells1080+2160+1080+1080+1080
Faces4320+4320+2160
Edges6480+720
Vertices1440
Vertex figure ( )v{3}x{3}
Petrie polygon Dodecagon
Coxeter group E6, [[3,3<sup>2,2</sup>]], 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.

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 symbol2r(122)
Coxeter-Dynkin diagram
or
5-faces126
4-faces2286
Cells10800
Faces19440
Edges12960
Vertices2160
Vertex figure
Coxeter group E6, [[3,3<sup>2,2</sup>]], order 103680
Properties convex

### Alternate names

• Bicantellated 221
• Birectified pentacontitetrapeton (barm) (Jonathan Bowers) [9]

### 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]

## Trirectified 1_22 polytope

Trirectified 122 polytope
Type Uniform 6-polytope
Schläfli symbol 3r{3,32,2}
Coxeter symbol3r(122)
Coxeter-Dynkin diagram
or
5-faces558
4-faces4608
Cells8640
Faces6480
Edges2160
Vertices270
Vertex figure
Coxeter group E6, [[3,3<sup>2,2</sup>]], order 103680
Properties convex

### Alternate names

• Tricantellated 221
• Trirectified pentacontitetrapeton (trim or cacam) (Jonathan Bowers) [10]

## Notes

1. Elte, 1912
2. Klitzing, (o3o3o3o3o *c3x - mo)
3. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 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* Lattices Archived 2016-01-30 at the Wayback Machine , Edward Pervin
6. Klitzing, (o3o3x3o3o *c3o - ram)
7. Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
8. Klitzing, Richard. "6D convex uniform polypeta o3o3x3o3o *c3o - ram".
9. Klitzing, (o3x3o3x3o *c3o - barm)
10. Klitzing, (x3o3o3o3x *c3o - cacam

## Related Research Articles

In geometry, a uniform polychoron is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In four-dimensional geometry, the rectified 5-cell is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams.

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

In 8-dimensional geometry, the 241 is a uniform 8-polytope, constructed within the symmetry of the E8 group.

In 6-dimensional geometry, the 221 polytope is a uniform 6-polytope, constructed within the symmetry of the E6 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 6-ic semi-regular figure. It is also called the Schläfli polytope.

In 7-dimensional geometry, the 321 polytope is a uniform 7-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 7-ic semi-regular figure.

In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group. It was discovered by Thorold Gosset, published in his 1900 paper. He called it an 8-ic semi-regular figure.

In geometry, the 222 honeycomb is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,32,2}. It is constructed from 221 facets and has a 122 vertex figure, with 54 221 polytopes around every vertex.

In 7-dimensional geometry, 133 is a uniform honeycomb, also given by Schläfli symbol {3,33,3}, and is composed of 132 facets.

In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.

In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

In five-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.

In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

## References

Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds