Dual uniform polyhedron
A dual uniform polyhedron is the dual of a uniform polyhedron. Where a uniform polyhedron is vertex-transitive, a dual uniform polyhedron is face-transitive. Enumeration The face-transitive polyhedra comprise the set of 9 regular polyhedra, two finite sets comprising 66 non-regular polyhedra, and two infinite sets: 5 regular convex Platonic solids, which are dual to each other (the regular tetrahedron is its own dual). 4 regular star Kepler-Poinsot solids, which are dual to each other. 13 convex Catalan solids, which are dual to the uniform convex Archimedean solids. 53 star polyhedra, which are dual to the uniform star polyhedra. The infinite series of bipyramids, which are dual to the uniform prisms, both convex and star. The infinite series of trapezohedra, which are dual to the uniform antiprisms, both convex and star. The full set are described by Wenninger, together with instructions for constructing models, in his book Dual Models. Dorman Luke construction For a uniform polyhedron, each face of the dual polyhedron may be derived from the original polyhedron's corresponding vertex figure by using the Dorman Luke construction. As an example, the illustration below shows the vertex figure (red) of the cuboctahedron being used to derive the corresponding face (blue) of