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.