Here's a
proof that the five
Platonic solids or
regular polyhedra are the only ones.
Suppose that f faces of the solid meet at each vertex
and that each face is a regular t-gon. Of course f
and t have to be at least 3. The sum of the angles at a
vertex is <2pi and each of these angles is (t-2)pi/t
(the angle of a regular t-gon). So this gives an inequality:
(f(t-2)pi)/t < 2pi
As a consequence
(f-2)(t-2) <4
It's obvious than any integer solutions to this with
f,t>= 3 must have either f or t
equal to 3. This gives us exactly the solutions:
(f,t)=(3,3),(3,4),(3,5),(4,3),(5,3)
These are, respectively, the
tetrahedron,
cube,
dodecahedron,
octahedron and
icosahedron.