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Problem 76: Equiprojective Polyhedra


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].



Partial and Related Results

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.

Related Open Problems

A generalization of the problem was posted on MathOverflow, 11Feb11: [O'R11]


Also in [CFG90], Problem B10.



Entry Revision History

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?, 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., 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.