The classification of Platonic solids is one of the first great classification theorems in mathematics. It's connected to the classification of finite subgroups of SO(3):
• The cyclic groups, ℤ/n
• The dihedral groups, Dn
• The symmetry group of the tetrahedron, A4
• The symmetry group of the cube or octahedron, S4
• The symmetry group of the dodecahedron or icosahedron, A5