Ununfoldable Polyhedra with Convex Faces
Abstract
Unfolding a convex polyhedron into a simple planar polygon is a wellstudied 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.
 Publication:

arXiv eprints
 Pub Date:
 August 1999
 arXiv:
 arXiv:cs/9908003
 Bibcode:
 1999cs........8003B
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Discrete Mathematics;
 G.2.1;
 F.2.2
 EPrint:
 14 pages, 9 figures, LaTeX 2e. To appear in Computational Geometry: Theory and Applications. Major revision with two new authors, solving the open problem about triangular faces