Ununfoldable polyhedra with convex faces

Authors:
Marshall Bern;Erik D. Demaine;David Eppstein;Eric Kuo;Andrea Mantler;Jack Snoeyink
Affiliations:
Xerox Palo Alto Research Center, 3333 Coyote Hill Rd., Palo Alto, CA;Department of Computer Science, University of Waterloo, Waterloo, ON N2L 3G1, Canada;Department of Information and Computer Science, University of California, Irvine, CA;EECS Computer Science Division, University of California, Berkeley, 387 Soda Hall #1776, Berkeley, CA;Department of Computer Science, University of British Columbia, Vancouver, BC V6T 1Z4, Canada;Department of Computer Science, University of British Columbia, Vancouver, BC V6T 1Z4, Canada and Department of Computer Science, University of North Carolina, Chapel Hill, NC
Venue:
Computational Geometry: Theory and Applications - Fourth CGC workshop on computional geometry
Year:
2003

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Abstract

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that "open" polyhedra with triangular faces may not be unfoldable no matter how they are cut.