1 _{22} | Rectified 1 _{22} | Birectified 1 _{22} |

2 _{21} | Rectified 2 _{21} | |

orthogonal projections in E_{6} Coxeter plane |
---|

In 6-dimensional geometry, the **1 _{22}** polytope is a uniform polytope, constructed from the E

- 1 22 polytope
- Alternate names
- Images
- Construction
- Related complex polyhedron
- Related polytopes and honeycomb
- Rectified 1 22 polytope
- Alternate names 2
- Images 2
- Construction 2
- Truncated 1 22 polytope
- Alternate names 3
- Construction 3
- Images 3
- Birectified 1 22 polytope
- Alternate names 4
- Images 4
- Trirectified 1 22 polytope
- Alternate names 5
- See also
- Notes
- References

Its Coxeter symbol is **1 _{22}**, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. There are two rectifications of the 1

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 | |
---|---|

Type | Uniform 6-polytope |

Family | 1_{k2} polytope |

Schläfli symbol | {3,3^{2,2}} |

Coxeter symbol | 1_{22} |

Coxeter-Dynkin diagram | or |

5-faces | 54: 27 1 _{21} 27 1 _{21} |

4-faces | 702: 270 1 _{11} 432 1 _{20} |

Cells | 2160: 1080 1 _{10} 1080 {3,3} |

Faces | 2160 {3} |

Edges | 720 |

Vertices | 72 |

Vertex figure | Birectified 5-simplex: 0 _{22} |

Petrie polygon | Dodecagon |

Coxeter group | E_{6}, [[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 E_{6}.

**Pentacontatetra-peton**(Acronym Mo) - 54-facetted polypeton (Jonathan Bowers)^{ [2] }

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) |

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, 1_{31}, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 5-simplex, 0_{22}, .

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

E_{6} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | k-figure | notes | |||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{5} | ( ) | f_{0} | 72 | 20 | 90 | 60 | 60 | 15 | 15 | 30 | 6 | 6 | r{3,3,3} | E_{6}/A_{5} = 72*6!/6! = 72 | |

A_{2}A_{2}A_{1} | { } | f_{1} | 2 | 720 | 9 | 9 | 9 | 3 | 3 | 9 | 3 | 3 | {3}×{3} | E_{6}/A_{2}A_{2}A_{1} = 72*6!/3!/3!/2 = 720 | |

A_{2}A_{1}A_{1} | {3} | f_{2} | 3 | 3 | 2160 | 2 | 2 | 1 | 1 | 4 | 2 | 2 | s{2,4} | E_{6}/A_{2}A_{1}A_{1} = 72*6!/3!/2/2 = 2160 | |

A_{3}A_{1} | {3,3} | f_{3} | 4 | 6 | 4 | 1080 | * | 1 | 0 | 2 | 2 | 1 | { }∨( ) | E_{6}/A_{3}A_{1} = 72*6!/4!/2 = 1080 | |

4 | 6 | 4 | * | 1080 | 0 | 1 | 2 | 1 | 2 | ||||||

A_{4}A_{1} | {3,3,3} | f_{4} | 5 | 10 | 10 | 5 | 0 | 216 | * | * | 2 | 0 | { } | E_{6}/A_{4}A_{1} = 72*6!/5!/2 = 216 | |

5 | 10 | 10 | 0 | 5 | * | 216 | * | 0 | 2 | ||||||

D_{4} | h{4,3,3} | 8 | 24 | 32 | 8 | 8 | * | * | 270 | 1 | 1 | E_{6}/D_{4} = 72*6!/8/4! = 270 | |||

D_{5} | h{4,3,3,3} | f_{5} | 16 | 80 | 160 | 80 | 40 | 16 | 0 | 10 | 27 | * | ( ) | E_{6}/D_{5} = 72*6!/16/5! = 27 | |

16 | 80 | 160 | 40 | 80 | 0 | 16 | 10 | * | 27 |

The regular complex polyhedron _{3}{3}_{3}{4}_{2}, , in has a real representation as the *1 _{22}* polytope in 4-dimensional space. It has 72 vertices, 216 3-edges, and 54 3{3}3 faces. Its complex reflection group is

Along with the semiregular polytope, ** 2 _{21} **, 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 | E_{3}=A_{2}A_{1} | E_{4}=A_{4} | E_{5}=D_{5} | E_{6} | E_{7} | E_{8} | E_{9} = = E_{8}^{+} | E_{10} = = E_{8}^{++} | |||

Coxeter diagram | |||||||||||

Symmetry (order) | [3^{−1,2,1}] | [3^{0,2,1}] | [3^{1,2,1}] | [[3<sup>2,2,1</sup>]] | [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} | 1_{02} | 1_{12} | 1_{22} | 1_{32} | 1_{42} | 1_{52} | 1_{62} |

The **1 _{22}** is related to the 24-cell by a geometric folding E6 → F4 of Coxeter-Dynkin diagrams, E6 corresponding to

E6/F4 Coxeter planes | |
---|---|

1_{22} | 24-cell |

D4/B4 Coxeter planes | |

1_{22} | 24-cell |

This polytope is the vertex figure for a uniform tessellation of 6-dimensional space, ** 2 _{22} **, .

Rectified 1_{22} | |
---|---|

Type | Uniform 6-polytope |

Schläfli symbol | 2r{3,3,3^{2,1}}r{3,3 ^{2,2}} |

Coxeter symbol | 0_{221} |

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 | E_{6}, [[3,3<sup>2,2</sup>]], order 103680 |

Properties | convex |

The **rectified 1 _{22}** polytope (also called

- Birectified 2
_{21}polytope - Rectified pentacontatetrapeton (acronym
*Ram*) - rectified 54-facetted polypeton (Jonathan Bowers)^{ [6] }

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

E6 [12] | D5 [8] | D4 / A2 [6] | B6 [12/2] |
---|---|---|---|

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

Its construction is based on the E_{6} 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: **t _{2}(2_{11})**, .

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] }

E_{6} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | k-figure | notes | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{2}A_{2}A_{1} | ( ) | f_{0} | 720 | 18 | 18 | 18 | 9 | 6 | 18 | 9 | 6 | 9 | 6 | 3 | 6 | 9 | 3 | 2 | 3 | 3 | {3}×{3}×{ } | E_{6}/A_{2}A_{2}A_{1} = 72*6!/3!/3!/2 = 720 | |

A_{1}A_{1}A_{1} | { } | f_{1} | 2 | 6480 | 2 | 2 | 1 | 1 | 4 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 1 | 1 | 2 | 2 | { }∨{ }∨( ) | E_{6}/A_{1}A_{1}A_{1} = 72*6!/2/2/2 = 6480 | |

A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 4320 | * | * | 1 | 2 | 1 | 0 | 0 | 2 | 1 | 1 | 2 | 0 | 1 | 2 | 1 | Sphenoid | E_{6}/A_{2}A_{1} = 72*6!/3!/2 = 4320 | |

3 | 3 | * | 4320 | * | 0 | 2 | 0 | 1 | 1 | 1 | 0 | 2 | 2 | 1 | 1 | 1 | 2 | ||||||

A_{2}A_{1}A_{1} | 3 | 3 | * | * | 2160 | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 1 | 0 | 2 | 2 | { }∨{ } | E_{6}/A_{2}A_{1}A_{1} = 72*6!/3!/2/2 = 2160 | |||

A_{2}A_{1} | {3,3} | f_{3} | 4 | 6 | 4 | 0 | 0 | 1080 | * | * | * | * | 2 | 1 | 0 | 0 | 0 | 1 | 2 | 0 | { }∨( ) | E_{6}/A_{2}A_{1} = 72*6!/3!/2 = 1080 | |

A_{3} | r{3,3} | 6 | 12 | 4 | 4 | 0 | * | 2160 | * | * | * | 1 | 0 | 1 | 1 | 0 | 1 | 1 | 1 | {3} | E_{6}/A_{3} = 72*6!/4! = 2160 | ||

A_{3}A_{1} | 6 | 12 | 4 | 0 | 4 | * | * | 1080 | * | * | 0 | 1 | 0 | 2 | 0 | 0 | 2 | 1 | { }∨( ) | E_{6}/A_{3}A_{1} = 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 | |||||

A_{4} | r{3,3,3} | f_{4} | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | 432 | * | * | * | * | 1 | 1 | 0 | { } | E_{6}/A_{4} = 72*6!/5! = 432 | |

A_{4}A_{1} | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 216 | * | * | * | 0 | 2 | 0 | E_{6}/A_{4}A_{1} = 72*6!/5!/2 = 216 | ||||

A_{4} | 10 | 30 | 10 | 20 | 0 | 0 | 5 | 0 | 5 | 0 | * | * | 432 | * | * | 1 | 0 | 1 | E_{6}/A_{4} = 72*6!/5! = 432 | ||||

D_{4} | h{4,3,3} | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 0 | 8 | * | * | * | 270 | * | 0 | 1 | 1 | E_{6}/D_{4} = 72*6!/8/4! = 270 | |||

A_{4}A_{1} | r{3,3,3} | 10 | 30 | 0 | 20 | 10 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 216 | 0 | 0 | 2 | E_{6}/A_{4}A_{1} = 72*6!/5!/2 = 216 | |||

A_{5} | 2r{3,3,3,3} | f_{5} | 20 | 90 | 60 | 60 | 0 | 15 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | 72 | * | * | ( ) | E_{6}/A_{5} = 72*6!/6! = 72 | |

D_{5} | rh{4,3,3,3} | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 0 | 40 | 16 | 16 | 0 | 10 | 0 | * | 27 | * | E_{6}/D_{5} = 72*6!/16/5! = 27 | |||

80 | 480 | 160 | 320 | 160 | 0 | 80 | 40 | 80 | 80 | 0 | 0 | 16 | 10 | 16 | * | * | 27 |

Truncated 1_{22} | |
---|---|

Type | Uniform 6-polytope |

Schläfli symbol | t{3,3^{2,2}} |

Coxeter symbol | t(1_{22}) |

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 | E_{6}, [[3,3<sup>2,2</sup>]], order 103680 |

Properties | convex |

- Truncated 1
_{22}polytope

Its construction is based on the E_{6} group and information can be extracted from the ringed Coxeter-Dynkin diagram representing this polytope: .

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

E6 [12] | D5 [8] | D4 / A2 [6] | B6 [12/2] |
---|---|---|---|

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

Birectified 1_{22} polytope | |
---|---|

Type | Uniform 6-polytope |

Schläfli symbol | 2r{3,3^{2,2}} |

Coxeter symbol | 2r(1_{22}) |

Coxeter-Dynkin diagram | or |

5-faces | 126 |

4-faces | 2286 |

Cells | 10800 |

Faces | 19440 |

Edges | 12960 |

Vertices | 2160 |

Vertex figure | |

Coxeter group | E_{6}, [[3,3<sup>2,2</sup>]], order 103680 |

Properties | convex |

- Bicantellated 2
_{21} - Birectified pentacontitetrapeton (barm) (Jonathan Bowers)
^{ [9] }

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

E6 [12] | D5 [8] | D4 / A2 [6] | B6 [12/2] |
---|---|---|---|

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

Trirectified 1_{22} polytope | |
---|---|

Type | Uniform 6-polytope |

Schläfli symbol | 3r{3,3^{2,2}} |

Coxeter symbol | 3r(1_{22}) |

Coxeter-Dynkin diagram | or |

5-faces | 558 |

4-faces | 4608 |

Cells | 8640 |

Faces | 6480 |

Edges | 2160 |

Vertices | 270 |

Vertex figure | |

Coxeter group | E_{6}, [[3,3<sup>2,2</sup>]], order 103680 |

Properties | convex |

- Tricantellated 2
_{21} - Trirectified pentacontitetrapeton (trim or cacam) (Jonathan Bowers)
^{ [10] }

- ↑ Elte, 1912
- ↑ Klitzing, (o3o3o3o3o *c3x - mo)
- ↑ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
- ↑ Coxeter, H. S. M.,
*Regular Complex Polytopes*, second edition, Cambridge University Press, (1991). p.30 and p.47 - ↑ The Voronoi Cells of the E6* and E7* Lattices Archived 2016-01-30 at the Wayback Machine , Edward Pervin
- ↑ Klitzing, (o3o3x3o3o *c3o - ram)
- ↑ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
- ↑ Klitzing, Richard. "6D convex uniform polypeta o3o3x3o3o *c3o - ram".
- ↑ Klitzing, (o3x3o3x3o *c3o - barm)
- ↑ Klitzing, (x3o3o3o3x *c3o - cacam

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, **2 _{31}** is a uniform polytope, constructed from the E7 group.

In 7-dimensional geometry, **1 _{32}** is a uniform polytope, constructed from the E7 group.

In 8-dimensional geometry, the **1 _{42}** is a uniform 8-polytope, constructed within the symmetry of the E

In 8-dimensional geometry, the **2 _{41}** is a uniform 8-polytope, constructed within the symmetry of the E

In 6-dimensional geometry, the **2 _{21}** polytope is a uniform 6-polytope, constructed within the symmetry of the E

In 7-dimensional geometry, the **3 _{21}** polytope is a uniform 7-polytope, constructed within the symmetry of the E

In 8-dimensional geometry, the **4 _{21}** is a semiregular uniform 8-polytope, constructed within the symmetry of the E

In geometry, the **2 _{22} honeycomb** is a uniform tessellation of the six-dimensional Euclidean space. It can be represented by the Schläfli symbol {3,3,3

In 7-dimensional geometry, **1 _{33}** is a uniform honeycomb, also given by Schläfli symbol {3,3

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.

- 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- (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 1_{22})

- (Paper 24) H.S.M. Coxeter,
- Klitzing, Richard. "6D uniform polytopes (polypeta)". o3o3o3o3o *c3x - mo, o3o3x3o3o *c3o - ram, o3x3o3x3o *c3o - barm

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