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 | ||||||||||||||||
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Family n | n-simplex | n-hypercube | n-orthoplex | n-demicube | 1k2 | 2k1 | k21 | pentagonal polytope | ||||||||
Group | An | Bn |
|
| Hn | |||||||||||
2 | Triangle | Square | p-gon (example: p=7) | Hexagon | Pentagon | |||||||||||
3 | Tetrahedron | Cube | Octahedron | Tetrahedron | Dodecahedron | Icosahedron | ||||||||||
4 | 5-cell | 16-cell | 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 |
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
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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 polychoron | Pentachoron | 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 |