Problem 76: Equiprojective Polyhedra

Statement

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.

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 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]

Appearances

Also in [CFG90], Problem B10.

Categories

polyhedra

Entry Revision History

J. O'Rourke, 31 Dec. 2010; 11 Feb 2011.

Bibliography

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.
Springer-Verlag, 1990.

HHLO+10

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.

HL08

Masud Hasan and Anna Lubiw.
Equiprojective polyhedra.
Comput. Geom. Th. Appl., 40(2):148-155, 2008.

She68

Geoffrey C. Shephard.
Twenty problems on convex polyhedra--II.
Math. Gaz., 52:359-367, 1968.