Classification of Convex Polyhedra whose Faces are Regular Polygons

Theorem
The convex polyhedra whose faces are all regular polygons are as follows:


 * The $5$ Platonic solids
 * The regular prisms and regular antiprisms, countably infinite in number
 * The $13$ Archimedean polyhedra
 * The $92$ Johnson polyhedra.

Proof
The Platonic solids are the convex polyhedra all of whose faces are congruent and whose vertices are regular.

From Five Platonic Solids, there are $5$ of these.

The regular prisms are made from two regular polygons of an arbitrary number of sides forming the bases, separated by as many squares as there are sides of the two bases, forming the lateral faces.

From Regular Prisms are Countably Infinite, they form a countably infinite set.

Note that when the bases are themselves squares, the regular prism is then a cube, and so has already been counted amongst the Platonic solids.

The regular antiprisms are made from two regular polygons of an arbitrary number of sides forming the bases, separated by twice as many equilateral triangles as there are sides of the two bases, forming the lateral faces.

From Regular Antiprisms are Countably Infinite, they form a countably infinite set.

Note that when the bases are themselves equilateral triangles, the regular antiprism is then a regular octahedron, and so has already been counted amongst the Platonic solids.

The Archimedean polyhedra consist of all vertex-transitive convex polyhedra whose faces are all regular polygons, excluding:
 * the Platonic solids
 * the regular prisms
 * the regular antiprisms.

From Thirteen Archimedean Polyhedra, there are $13$ of these.

The Johnson polyhedra are all the other convex polyhedra whose faces are all regular polygons.

From 92 Johnson Polyhedra, there are $92$ of these.