Nine Regular Polyhedra

Theorem
There exist $9$ regular polyhedra.

Proof
From Five Platonic Solids, there exist $5$ regular polyhedra which are convex:
 * the regular tetrahedron
 * the cube
 * the regular octahedron
 * the regular dodecahedron
 * the regular icosahedron.

From Four Kepler-Poinsot Polyhedra, there are $4$ regular polyhedra which are non-convex:

making the total $9$.