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Identify or construct all k-equiprojective polyhedra. A polyhedron P is k-equiprojective if its orthogonal projection to a plane is a k-gon in every direction not parallel to a face of P. Thus a cube is 6-equiprojective.
Geoffrey Shephard in [She68].
A characterization is detailed in [HL08]: “A polyhedron is equiprojective iff its set of edge-face pairs can be partitioned into compensating pairs.” For term definitions, see the original paper. Building on this work, a recent paper [HHLO+10] establishes that any equiprojective polyhedron has at least one pair of parallel faces, that there is no 3- or 4-equiprojective polyhedron, and the triangular prism is the only 5-equiprojective polyhedron.
A generalization of the problem was posted on MathOverflow, 11Feb11: [O'R11]
Also in [CFG90], Problem B10.
J. O’Rourke, 31 Dec. 2010; 11 Feb 2011.
Joseph O’Rourke. What is determined by the combinatorics of the shadows of a convex polyhedron? http://mathoverflow.net/questions/55124/, February 2011.
H. P. Croft, K. J. Falconer, and R. K. Guy. Unsolved Problems in Geometry. Springer-Verlag, 1990.
Masud Hasan, Mohammad Houssain, Alejandro Lopez-Oritz, Sabrina Nusrat, Saad Quader, and Nabila Rahman. Some new equiprojective polyhedra. http://arxiv.org/abs/1009.2252, 2010.
Masud Hasan and Anna Lubiw. Equiprojective polyhedra. Comput. Geom. Th. Appl., 40(2):148–155, 2008.
Geoffrey C. Shephard. Twenty problems on convex polyhedra—II. Math. Gaz., 52:359–367, 1968.