Polytope families

There are several families of symmetric polytopes with irreducible symmetry which have a member in more than one dimensionality. These are tabulated here with Petrie polygon projection graphs and Coxeter-Dynkin diagrams.

Table of irreducible polytope families

Family
n  | n-simplex | n-hypercube | n-orthoplex  | n-demicube | 1k2 | 2k1 | k21 | pentagonal polytope
Group | An | Bn |  | I2(p)  | Dn

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| E6 | E7 | E8 | F4 | G2

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|
|
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2 |

Triangle

Square

p-gon
(example: p=7)  |

Hexagon |

Pentagon
3 |

Tetrahedron |

Cube |

Octahedron |

Tetrahedron |  |

Dodecahedron |

Icosahedron
4  |

5-cell |

Tesseract

16-cell |

Demitesseract

24-cell |

120-cell |

600-cell
5 |

5-simplex |

5-cube |

5-orthoplex |

5-demicube |  |  |
6 |

6-simplex |

6-cube |

6-orthoplex |

6-demicube |

122 |

221 |  |
7  |

7-simplex |

7-cube |

7-orthoplex |

7-demicube |

132 |

231 |

321 |  |
8  |

8-simplex  |

8-cube  |

8-orthoplex  |

8-demicube  |

142 |

241 |

421 |  |
9  |

9-simplex |

9-cube |

9-orthoplex |

9-demicube |
10  |

10-simplex |

10-cube |

10-orthoplex |

10-demicube |

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Fundamental convex regular and uniform polytopes in dimensions 2–10

Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube |  | Dodecahedron • Icosahedron
Uniform 4-polytope | 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 |  |
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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