On fixed size projection of simplicial polyhedra

• We consider the open problem of enumerating all equiprojective polyhedra.
• We show that simplicial polyhedra cannot be equiprojective.
• We extend the idea of equiprojectivity to biprojectivity.
• We show that some simplicial polyhedra are not (k,k+1)(k,k+1)-biprojective.
• Results are based on the characterization of equiprojective polyhedra published earlier.

A convex polyhedron P is equiprojective (similarly, biprojective) if, for some fixed k   (similarly, k1k1 and k2k2), the orthogonal projection (or “shadow”) of P in every direction, except directions parallel to faces of P, is a k  -gon (similarly, k1k1- or k2k2-gon). Since 1968, it is an open problem to construct all equiprojective polyhedra, while the only results include a characterization, a recognition algorithm, and some non-trivial examples of equiprojective polyhedra. In this note, we show that simplicial polyhedra cannot be equiprojective. Then, we extend the idea of equiprojectivity to biprojectivity and show that simplicial polyhedra having all faces in parallel pairs are not (k,k+1)(k,k+1)-biprojective.