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Problem 76: Equiprojective Polyhedra
 Statement
 Identify or construct all kequiprojective polyhedra.
A polyhedron P is kequiprojective if its orthogonal
projection to a plane is a kgon in every direction not parallel
to a face of P.
Thus a cube is 6equiprojective.
 Origin
 Geoffrey Shephard in
[She68].
 Status/Conjectures
 Open.
 Partial and Related Results
 A characterization is detailed in [HL08]:
``A polyhedron is equiprojective iff its set of edgeface 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 4equiprojective polyhedron,
and the triangular prism is the only 5equiprojective polyhedron.
 Related Open Problems
 A generalization of
the problem was posted on MathOverflow, 11Feb11:
[O'R11]
 Appearances
 Also in [CFG90], Problem B10.
 Categories
 polyhedra
 Entry Revision History
 J. O'Rourke, 31 Dec. 2010; 11 Feb 2011.
 O'R11

Joseph O'Rourke.
What is determined by the combinatorics of the shadows of a convex
polyhedron?
http://mathoverflow.net/questions/55124/, February 2011.
 CFG90

H. P. Croft, K. J. Falconer, and R. K. Guy.
Unsolved Problems in Geometry.
SpringerVerlag, 1990.
 HHLO+10

Masud Hasan, Mohammad Houssain, Alejandro LopezOritz, Sabrina Nusrat, Saad
Quader, and Nabila Rahman.
Some new equiprojective polyhedra.
http://arxiv.org/abs/1009.2252, 2010.
 HL08

Masud Hasan and Anna Lubiw.
Equiprojective polyhedra.
Comput. Geom. Th. Appl., 40(2):148155, 2008.
 She68

Geoffrey C. Shephard.
Twenty problems on convex polyhedraII.
Math. Gaz., 52:359367, 1968.
The Open Problems Project  December 04, 2015