A "regular polyheron", also called a "Platonic solid", is a solid figure with all faces the same polygon, all edges the same length, all angles the same.
If you mean "an infinite number of different platonic solids", that is, different numbers of sides, etc., then, far from being an infinite number of them, there are only five of them:
Tetrahedron: Four faces each being an equilateral triangle. Four faces, six edges, four vertices.
Hexahedron (cube): Six faces each being a square. Six faces, twelve edges, eight vertices.
Octahedron: Eight faces, each being an equilateral triangle. Eight faces, twelve edges, four vertices.
Dodecahedron: Twelve faces, each being a regular pentagon. Twelve faces, thirty edges, twenty vertices.
Icosahedron: twenty faces, each being an equilateral triangle. Twenty faces, thirty edges, twenty vertices.
Notice that all of these satisfies "Euler's formula": faces- edges+ vertices= 2.
Also they come in pairs or "duals", swapping number of faces with number of vertices: If you were to mark the center point of each face and then connect those points, the result would be the "dual" polyhedron. The hexahedron is dual to the octahedron, the dodecahedron is dual to the icosahedron and the tetrahedron is dual to itself.
You can see pictures of them here:
Platonic Solid -- from Wolfram MathWorld