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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
- 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
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:
- Also in [CFG90], Problem B10.
- Entry Revision History
- J. O'Rourke, 31 Dec. 2010; 11 Feb 2011.
What is determined by the combinatorics of the shadows of a convex
http://mathoverflow.net/questions/55124/, February 2011.
H. P. Croft, K. J. Falconer, and R. K. Guy.
Unsolved Problems in Geometry.
Masud Hasan, Mohammad Houssain, Alejandro Lopez-Oritz, Sabrina Nusrat, Saad
Quader, and Nabila Rahman.
Some new equiprojective polyhedra.
Masud Hasan and Anna Lubiw.
Comput. Geom. Th. Appl., 40(2):148-155, 2008.
Geoffrey C. Shephard.
Twenty problems on convex polyhedra--II.
Math. Gaz., 52:359-367, 1968.
The Open Problems Project - September 19, 2017