regular polyhedron

A convex polyhedron whose faces are all identical regular polygons, and whose dihedral angles are all identical and whose solid angles are all identical. Only five exist, the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. These are also called the platonic solids.

The convexity constraint is equivalent to requiring the polyhedron be non-self-intersecting. A polyhedron which is self-intersecting but otherwise meets the constraints (except possibly for self-intersecting faces as well) is called a stellated polyhedron.

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.