Thirteen Catalan Polyhedra

Theorem
There exist exactly $13$ distinct Catalan polyhedra:


 * Triakis tetrahedron
 * Triakis octahedron
 * Disdyakis dodecahedron
 * Tetrakis hexahedron
 * Triakis icosahedron
 * Disdyakis triacontahedron
 * Pentakis dodecahedron
 * Rhombic dodecahedron
 * Rhombic triacontahedron
 * Deltoidal icositetrahedron
 * Deltoidal hexecontahedron
 * Pentagonal icositetrahedron
 * Pentagonal hexecontahedron

Proof
By definition, the Catalan polyhedra are the dual polyhedra of the Archimedean polyhedra.

There are $13$ Archimedean polyhedra, and so there are $13$ Catalan polyhedra.