Polytope families
# 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 | $1_{k2}$ | $2_{k1}$ | $k_{21}$ | pentagonal polytope | pentagonal polytope |
| - | - | - | - | - | - | - | - | - | - |
| Group | $A_n$ | $B_n$ | $B_n$ | $I_2(p)$ $D_n$ | $E_6$ $E_7$ $E_8$ $F_4$ $G_2$ | $E_6$ $E_7$ $E_8$ $F_4$ $G_2$ | $E_6$ $E_7$ $E_8$ $F_4$ $G_2$ | $H_n$ | $H_n$ |
| 2 | Triangle | Square | Square | p-gon (example: p=7) | Hexagon | Hexagon | Hexagon | Pentagon | Pentagon |
| 3 | Tetrahedron | Cube | Octahedron | Tetrahedron | | | | Dodecahedron | Icosahedron |
| 4 | 5-cell | Tesseract | 16-cell | Demitesseract | 24-cell | 24-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 | $1_{22}$ | $2_{21}$ | $2_{21}$ | | |
| 7 | 7-simplex | 7-cube | 7-orthoplex | 7-demicube | $1_{32}$ | $2_{31}$ | $3_{21}$ | | |
| 8 | 8-simplex | 8-cube | 8-orthoplex | 8-demicube | $1_{42}$ | $2_{41}$ | $4_{21}$ | | |
| 9 | 9-simplex | 9-cube | 9-orthoplex | 9-demicube | | | | | |
| 10 | 10-simplex | 10-cube | 10-orthoplex | 10-demicube | | | | | |

vteFundamental convex regular and uniform polytopes in dimensions 2–10 :

| Family | $A_n$ | $B_n$ | $I_2(p)$ / $D_n$ | $E_6$ / $E_7$ / $E_8$ / $F_4$ / $G_2$ | $H_n$ |
| - | - | - | - | - | - |
| 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 | $1_{22}$ • $2_{21}$ | |
| Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | $1_{32}$ • $2_{31}$ • $3_{21}$ | |
| Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | $1_{42}$ • $2_{41}$ • $4_{21}$ | |
| 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 | $1_{k2}$ • $2_{k1}$ • $k_{21}$ | n-pentagonal polytope |
| Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |||||||||||||
