Smallest and some new equiprojective polyhedra

A convex polyhedron P is k-equiprojective if for all of its orthogonal projections, except those parallel to the faces of P, the number of vertices in the shadow boundary is k. Finding an algorithm to construct all equiprojective polyhedra is an open problem first posed in 1968. In this paper we give lower bounds on the value of k and the size of an equiprojective polyhedron. We prove that there is no 3- or 4-equiprojective polyhedra and a triangular prism is the only 5-equiprojective polyhedron. We also discover some new equiprojective polyhedra.