1 Smallest and Some New Equiprojective Polyhedra Masud Hasan 1 , Mohammad Monoar Hossain 1 , Sabrina Nusrat 1 and Alejandro Lopez-Ortiz 2 1 Department of Computer Science and Engineering Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh 2 Cheriton School of Computer Science, University of Waterloo, Ontario N2L 3G1, Canada Email: masudhasan@cse.buet.ac.bd, alopez-o@uwaterloo.ca Abstract— 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. Keywords: Algorithm, equiprojective polyhedra, John- son solid, lower bound, orthogonal projection I. I NTRODUCTION The problem of constructing new polyhedra based on certain mathematical properties has been extensively studied since antiquity in the fields of architecture, art, ornament, nature, cartography, and even in philosophy and literature. (See the book [3] and the web page [6] for some interesting discussions on the history of dis- covering new polyhedra.) Regularity of faces, edges and vertices, and symmetry are some popular criteria for constructing new polyhedra. For example, in a platonic solid (platonic solids are the most primitive convex polyhedra) each vertex is incident to the same number of identical regular faces [3], [6]. A convex polyhedron P is k-equiprojective if its shadow (i.e., the boundary of orthogonal projection) is a k-gon in every direction, except directions parallel to faces of P . A cube is 6-equiprojective, a triangular prism is 5-equiprojective (in fact, any n-gonal prism is (n + 2)-equiprojective [10], [11]), and a tetrahedron is not equiprojective. See Figure 1. In 1968, in a paper [13] entitled, “Twenty problems on convex poly- hedra,” Shephard defined equiprojective polyhedra, gave the examples above, and asked for a method to construct all equiprojective polyhedra. Later, Croft, Falconer, and Guy included this problem in their book “Unsolved Problems in Geometry” [2]. (a) (b) Fig. 1. (a) Some equiprojective polyhedra, and (b) a non- equiprojective polyhedron. So far little progress has been made in this problem. Hasan and Lubiw [10], [11] gave a new characteriza- tion and an O(n log n) time recognition algorithm for equiprojective polyhedra. Their characterization shows that the class of equiprojective polyhedra is rich. For example, zonohedra, which is an infinite class of convex polyhedra, are equiprojective [10], [11]. Rahman et. al. [12] also proposed some new non- trivial equiprojective polyhedra, which they achieved by cutting and gluing some existing convex polyhedra. In other related work, Hasan, in his PhD thesis [7], extended the idea of equiprojectivity from the number of vertices in the shadow to the number of visible faces, visible edges, and visible vertices in the projec- tion. However, the original problem of constructing all equiprojective polyhedra still remains open. In this paper we further explore the class of equipro- jective polyhedra. As a lower bound on smallest equiprojective polyhedra, we prove that there is no 3- or 4-equiprojective polyhedra and a triangular prism is the only 5-equiprojective polyhedron (Section 3). We also discover some new equiprojective polyhedra by cutting some existing polyhedra (Section 4). Most of our newly discovered equiprojective polyhedra are non-trivial and to our knowledge, none of them has been discovered before as an equiprojective polyhedron. II. PRELIMINARIES For rest of the paper by polyhedra we mean convex polyhedra and by projection (shadow) of a polyhe-