Four Kepler-Poinsot Polyhedra

Theorem
There exist exactly four Kepler-Poinsot polyhedra:
 * $(1): \quad$ the small stellated dodecahedron
 * $(2): \quad$ the great stellated dodecahedron
 * $(3): \quad$ the great godecahedron
 * $(4): \quad$ the great icosahedron