Classification theorems in mathematics always have the form: "Any X belongs either to one of n infinite series of classical objects, or is one of the m exceptions". For example:
- A regular polyhedron in R^n is either a regular simplex, cube, or cocube, or it is an isocahedron, a dodecahedron, a 24-cell, a 120-cell, or a 600-cell.
- A simple Lie algebra is either sl_n, so_n, sp_2n, or a g_2, f_4, e_6, e_7, e_8
- A finite simple group is either alternating, a finite group of Lie type, or one of the 26 sporadic groups.
These exceptions are often overlooked, but in fact, there is a system behind the madness which is almost as strong as the uniformity in the classical case. I will show how the existence of the icosahedron naturally leads to the exceptional isomorphisms PSL(2,5)=A_5, PSL(2,9)=A_6, which naturally lead to the existence of the sporadic Mathieu group, which naturally leads to the existence of the binary Golay code, which naturally leads to the existence of the Leech lattice, which naturally leads to the existence of the Monster group.