3 _{21} | 2 _{31} | 1_{32} | |||

Rectified 3 _{21} | birectified 3 _{21} | ||||

Rectified 2 _{31} | Rectified 1 _{32} | ||||

Orthogonal projections in E_{7} Coxeter plane |
---|

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

- 1 32 polytope
- Alternate names
- Images
- Construction
- Related polytopes and honeycombs
- Rectified 1 32 polytope
- Alternate names 2
- Construction 2
- Images 2
- See also
- Notes
- References

Its Coxeter symbol is **1 _{32}**, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.

The **rectified 1 _{32}** is constructed by points at the mid-edges of the

These polytopes are part of a family of 127 (2^{7}-1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

1_{32} | |
---|---|

Type | Uniform 7-polytope |

Family | 1_{k2} polytope |

Schläfli symbol | {3,3^{3,2}} |

Coxeter symbol | 1_{32} |

Coxeter diagram | |

6-faces | 182: 56 1 _{22} 126 1 _{31} |

5-faces | 4284: 756 1 _{21} 1512 1 _{21} 2016 {3 ^{4}} |

4-faces | 23688: 4032 {3 ^{3}} 7560 1 _{11} 12096 {3 ^{3}} |

Cells | 50400: 20160 {3 ^{2}} 30240 {3 ^{2}} |

Faces | 40320 {3} |

Edges | 10080 |

Vertices | 576 |

Vertex figure | t_{2}{3^{5}} |

Petrie polygon | Octadecagon |

Coxeter group | E_{7}, [3^{3,2,1}], order 2903040 |

Properties | convex |

This polytope can tessellate 7-dimensional space, with symbol **1 _{33}**, and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual E

- Emanuel Lodewijk Elte named it V
_{576}(for its 576 vertices) in his 1912 listing of semiregular polytopes.^{ [2] } - Coxeter called it
**1**for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch._{32} *Pentacontihexa-hecatonicosihexa-exon*(Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers)^{ [3] }

E7 | E6 / F4 | B7 / A6 |
---|---|---|

[18] | [12] | [7x2] |

A5 | D7 / B6 | D6 / B5 |

[6] | [12/2] | [10] |

D5 / B4 / A4 | D4 / B3 / A2 / G2 | D3 / B2 / A3 |

[8] | [6] | [4] |

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space.

The facet information can be extracted from its Coxeter-Dynkin diagram,

Removing the node on the end of the 2-length branch leaves the 6-demicube, 1_{31},

Removing the node on the end of the 3-length branch leaves the 1_{22},

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 0_{32},

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{ [4] }

E_{7} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | k-figures | notes | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{6} | ( ) | f_{0} | 576 | 35 | 210 | 140 | 210 | 35 | 105 | 105 | 21 | 42 | 21 | 7 | 7 | 2r{3,3,3,3,3} | E_{7}/A_{6} = 72*8!/7! = 576 | |

A_{3}A_{2}A_{1} | { } | f_{1} | 2 | 10080 | 12 | 12 | 18 | 4 | 12 | 12 | 6 | 12 | 3 | 4 | 3 | {3,3}x{3} | E_{7}/A_{3}A_{2}A_{1} = 72*8!/4!/3!/2 = 10080 | |

A_{2}A_{2}A_{1} | {3} | f_{2} | 3 | 3 | 40320 | 2 | 3 | 1 | 6 | 3 | 3 | 6 | 1 | 3 | 2 | { }∨{3} | E_{7}/A_{2}A_{2}A_{1} = 72*8!/3!/3!/2 = 40320 | |

A_{3}A_{2} | {3,3} | f_{3} | 4 | 6 | 4 | 20160 | * | 1 | 3 | 0 | 3 | 3 | 0 | 3 | 1 | {3}∨( ) | E_{7}/A_{3}A_{2} = 72*8!/4!/3! = 20160 | |

A_{3}A_{1}A_{1} | 4 | 6 | 4 | * | 30240 | 0 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | Phyllic disphenoid | E_{7}/A_{3}A_{1}A_{1} = 72*8!/4!/2/2 = 30240 | |||

A_{4}A_{2} | {3,3,3} | f_{4} | 5 | 10 | 10 | 5 | 0 | 4032 | * | * | 3 | 0 | 0 | 3 | 0 | {3} | E_{7}/A_{4}A_{2} = 72*8!/5!/3! = 4032 | |

D_{4}A_{1} | {3,3,4} | 8 | 24 | 32 | 8 | 8 | * | 7560 | * | 1 | 2 | 0 | 2 | 1 | { }∨( ) | E_{7}/D_{4}A_{1} = 72*8!/8/4!/2 = 7560 | ||

A_{4}A_{1} | {3,3,3} | 5 | 10 | 10 | 0 | 5 | * | * | 12096 | 0 | 2 | 1 | 1 | 2 | E_{7}/A_{4}A_{1} = 72*8!/5!/2 = 12096 | |||

D_{5}A_{1} | h{4,3,3,3} | f_{5} | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 0 | 756 | * | * | 2 | 0 | { } | E_{7}/D_{5}A_{1} = 72*8!/16/5!/2 = 756 | |

D_{5} | 16 | 80 | 160 | 40 | 80 | 0 | 10 | 16 | * | 1512 | * | 1 | 1 | E_{7}/D_{5} = 72*8!/16/5! = 1512 | ||||

A_{5}A_{1} | {3,3,3,3,3} | 6 | 15 | 20 | 0 | 15 | 0 | 0 | 6 | * | * | 2016 | 0 | 2 | E_{7}/A_{5}A_{1} = 72*8!/6!/2 = 2016 | |||

E_{6} | {3,3^{2,2}} | f_{6} | 72 | 720 | 2160 | 1080 | 1080 | 216 | 270 | 216 | 27 | 27 | 0 | 56 | * | ( ) | E_{7}/E_{6} = 72*8!/72/6! = 56 | |

D_{6} | h{4,3,3,3,3} | 32 | 240 | 640 | 160 | 480 | 0 | 60 | 192 | 0 | 12 | 32 | * | 126 | E_{7}/D_{6} = 72*8!/32/6! = 126 |

The 1_{32} is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. The next figure is the Euclidean honeycomb 1_{33} and the final is a noncompact hyperbolic honeycomb, 1_{34}.

Space | Finite | Euclidean | Hyperbolic | |||
---|---|---|---|---|---|---|

n | 4 | 5 | 6 | 7 | 8 | 9 |

Coxeter group | A_{3}A_{1} | A_{5} | D_{6} | E_{7} | =E_{7}^{+} | =E_{7}^{++} |

Coxeter diagram | ||||||

Symmetry | [3^{−1,3,1}] | [3^{0,3,1}] | [3^{1,3,1}] | [3^{2,3,1}] | [[3<sup>3,3,1</sup>]] | [3^{4,3,1}] |

Order | 48 | 720 | 23,040 | 2,903,040 | ∞ | |

Graph | - | - | ||||

Name | 1_{3,-1} | 1_{30} | 1_{31} | 1_{32} | 1_{33} | 1_{34} |

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} |

Rectified 1_{32} | |
---|---|

Type | Uniform 7-polytope |

Schläfli symbol | t_{1}{3,3^{3,2}} |

Coxeter symbol | 0_{321} |

Coxeter-Dynkin diagram | |

6-faces | 758 |

5-faces | 12348 |

4-faces | 72072 |

Cells | 191520 |

Faces | 241920 |

Edges | 120960 |

Vertices | 10080 |

Vertex figure | {3,3}×{3}×{} |

Coxeter group | E_{7}, [3^{3,2,1}], order 2903040 |

Properties | convex |

The **rectified 1 _{32}** (also called

- Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym rolin) (Jonathan Bowers)
^{ [5] }

It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, , and the ring represents the position of the active mirror(s).

Removing the node on the end of the 3-length branch leaves the rectified 1_{22} polytope,

Removing the node on the end of the 2-length branch leaves the demihexeract, 1_{31},

Removing the node on the end of the 1-length branch leaves the birectified 6-simplex,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×{},

Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.^{ [6] }

E_{7} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | k-figures | notes | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

A_{3}A_{2}A_{1} | ( ) | f_{0} | 10080 | 24 | 24 | 12 | 36 | 8 | 12 | 36 | 18 | 24 | 4 | 12 | 18 | 24 | 12 | 6 | 6 | 8 | 12 | 6 | 3 | 4 | 2 | 3 | {3,3}x{3}x{ } | E_{7}/A_{3}A_{2}A_{1} = 72*8!/4!/3!/2 = 10080 | |

A_{2}A_{1}A_{1} | { } | f_{1} | 2 | 120960 | 2 | 1 | 3 | 1 | 2 | 6 | 3 | 3 | 1 | 3 | 6 | 6 | 3 | 1 | 3 | 3 | 6 | 2 | 1 | 3 | 1 | 2 | ( )v{3}v{ } | E_{7}/A_{2}A_{1}A_{1} = 72*8!/3!/2/2 = 120960 | |

A_{2}A_{2} | 0_{1} | f_{2} | 3 | 3 | 80640 | * | * | 1 | 1 | 3 | 0 | 0 | 1 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 1 | 0 | 3 | 1 | 1 | {3}v( )v( ) | E_{7}/A_{2}A_{2} = 72*8!/3!/3! = 80640 | |

A_{2}A_{2}A_{1} | 3 | 3 | * | 40320 | * | 0 | 2 | 0 | 3 | 0 | 1 | 0 | 6 | 0 | 3 | 0 | 3 | 0 | 6 | 0 | 1 | 3 | 0 | 2 | {3}v{ } | E_{7}/A_{2}A_{2}A_{1} = 72*8!/3!/3!/2 = 40320 | |||

A_{2}A_{1}A_{1} | 3 | 3 | * | * | 120960 | 0 | 0 | 2 | 1 | 2 | 0 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | { }v{ }v( ) | E_{7}/A_{2}A_{1}A_{1} = 72*8!/3!/2/2 = 120960 | |||

A_{3}A_{2} | 0_{2} | f_{3} | 4 | 6 | 4 | 0 | 0 | 20160 | * | * | * | * | 1 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 3 | 1 | 0 | {3}v( ) | E_{7}/A_{3}A_{2} = 72*8!/4!/3! = 20160 | |

0_{11} | 6 | 12 | 4 | 4 | 0 | * | 20160 | * | * | * | 1 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | 3 | 0 | 0 | 3 | 0 | 1 | |||||

A_{3}A_{1} | 6 | 12 | 4 | 0 | 4 | * | * | 60480 | * | * | 0 | 1 | 1 | 2 | 0 | 0 | 1 | 2 | 2 | 1 | 0 | 2 | 1 | 1 | Sphenoid | E_{7}/A_{3}A_{1} = 72*8!/4!/2 = 60480 | |||

A_{3}A_{1}A_{1} | 6 | 12 | 0 | 4 | 4 | * | * | * | 30240 | * | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 0 | 1 | 2 | 0 | 2 | { }v{ } | E_{7}/A_{3}A_{1}A_{1} = 72*8!/4!/2/2 = 30240 | |||

A_{3}A_{1} | 0_{2} | 4 | 6 | 0 | 0 | 4 | * | * | * | * | 60480 | 0 | 0 | 0 | 2 | 1 | 1 | 0 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | Sphenoid | E_{7}/A_{3}A_{1} = 72*8!/4!/2 = 60480 | ||

A_{4}A_{2} | 0_{21} | f_{4} | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | 4032 | * | * | * | * | * | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | {3} | E_{7}/A_{4}A_{2} = 72*8!/5!/3! = 4032 | |

A_{4}A_{1} | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 12096 | * | * | * | * | 1 | 2 | 0 | 0 | 0 | 2 | 1 | 0 | { }v() | E_{7}/A_{4}A_{1} = 72*8!/5!/2 = 12096 | |||

D_{4}A_{1} | 0_{111} | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 8 | 0 | * | * | 7560 | * | * | * | 1 | 0 | 2 | 0 | 0 | 2 | 0 | 1 | E_{7}/D_{4}A_{1} = 72*8!/8/4!/2 = 7560 | |||

A_{4} | 0_{21} | 10 | 30 | 10 | 0 | 20 | 0 | 0 | 5 | 0 | 5 | * | * | * | 24192 | * | * | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | ( )v( )v( ) | E_{7}/A_{4} = 72*8!/5! = 34192 | ||

A_{4}A_{1} | 10 | 30 | 0 | 10 | 20 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 12096 | * | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | { }v() | E_{7}/A_{4}A_{1} = 72*8!/5!/2 = 12096 | |||

0_{3} | 5 | 10 | 0 | 0 | 10 | 0 | 0 | 0 | 0 | 5 | * | * | * | * | * | 12096 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | |||||

D_{5}A_{1} | 0_{211} | f_{5} | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 40 | 0 | 16 | 16 | 10 | 0 | 0 | 0 | 756 | * | * | * | * | 2 | 0 | 0 | { } | E_{7}/D_{5}A_{1} = 72*8!/16/5!/2 = 756 | |

A_{5} | 0_{22} | 20 | 90 | 60 | 0 | 60 | 15 | 0 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | * | 4032 | * | * | * | 1 | 1 | 0 | E_{7}/A_{5} = 72*8!/6! = 4032 | |||

D_{5} | 0_{211} | 80 | 480 | 160 | 160 | 320 | 0 | 40 | 80 | 80 | 80 | 0 | 0 | 10 | 16 | 16 | 0 | * | * | 1512 | * | * | 1 | 0 | 1 | E_{7}/D_{5} = 72*8!/16/5! = 1512 | |||

A_{5} | 0_{31} | 15 | 60 | 20 | 0 | 60 | 0 | 0 | 15 | 0 | 30 | 0 | 0 | 0 | 6 | 0 | 6 | * | * | * | 4032 | * | 0 | 1 | 1 | E_{7}/A_{5} = 72*8!/6! = 4032 | |||

A_{5}A_{1} | 15 | 60 | 0 | 20 | 60 | 0 | 0 | 0 | 15 | 30 | 0 | 0 | 0 | 0 | 6 | 6 | * | * | * | * | 2016 | 0 | 0 | 2 | E_{7}/A_{5}A_{1} = 72*8!/6!/2 = 2016 | ||||

E_{6} | 0_{221} | f_{6} | 720 | 6480 | 4320 | 2160 | 4320 | 1080 | 1080 | 2160 | 1080 | 1080 | 216 | 432 | 270 | 432 | 216 | 0 | 27 | 72 | 27 | 0 | 0 | 56 | * | * | ( ) | E_{7}/E_{6} = 72*8!/72/6! = 56 | |

A_{6} | 0_{32} | 35 | 210 | 140 | 0 | 210 | 35 | 0 | 105 | 0 | 105 | 0 | 21 | 0 | 42 | 0 | 21 | 0 | 7 | 0 | 7 | 0 | * | 576 | * | E_{7}/A_{6} = 72*8!/7! = 576 | |||

D_{6} | 0_{311} | 240 | 1920 | 640 | 640 | 1920 | 0 | 160 | 480 | 480 | 960 | 0 | 0 | 60 | 192 | 192 | 192 | 0 | 0 | 12 | 32 | 32 | * | * | 126 | E_{7}/D_{6} = 72*8!/32/6! = 126 |

E7 | E6 / F4 | B7 / A6 |
---|---|---|

[18] | [12] | [14] |

A5 | D7 / B6 | D6 / B5 |

[6] | [12/2] | [10] |

D5 / B4 / A4 | D4 / B3 / A2 / G2 | D3 / B2 / A3 |

[8] | [6] | [4] |

- ↑ The Voronoi Cells of the E
_{6}^{*}and E_{7}^{*}Lattices Archived 2016-01-30 at the Wayback Machine , Edward Pervin - ↑ Elte, 1912
- ↑ Klitzing, (o3o3o3x *c3o3o3o - lin)
- ↑ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203
- ↑ Klitzing, (o3o3x3o *c3o3o3o - rolin)
- ↑ Coxeter, Regular Polytopes, 11.8 Gossett figures in six, seven, and eight dimensions, p. 202-203

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

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 8-dimensional geometry, the **2 _{51}** honeycomb is a space-filling uniform tessellation. It is composed of 2

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, the **3 _{31} honeycomb** is a uniform honeycomb, also given by Schläfli symbol {3,3,3,3

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

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

In geometry, the **5 _{21} honeycomb** is a uniform tessellation of 8-dimensional Euclidean space. The symbol 5

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

In geometry, an **E _{9} honeycomb** is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E

In geometry, the **simplectic honeycomb** is a dimensional infinite series of honeycombs, based on the affine Coxeter group symmetry. It is given a Schläfli symbol {3^{[n+1]}}, and is represented by a Coxeter-Dynkin diagram as a cyclic graph of *n+1* nodes with one node ringed. It is composed of n-simplex facets, along with all rectified n-simplices. It can be thought of as an n-dimensional hypercubic honeycomb that has been subdivided along all hyperplanes , then stretched along its main diagonal until the simplices on the ends of the hypercubes become regular. The vertex figure of an *n-simplex honeycomb* is an expanded n-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]

- (Paper 24) H.S.M. Coxeter,
- Klitzing, Richard. "7D uniform polytopes (polyexa)". o3o3o3x *c3o3o3o - lin, o3o3x3o *c3o3o3o - rolin

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.