A classic open question in geometry is whether every convex polyhedron can be cut along its edges and flattened into the plane without any overlap. Such a collection of cuts is called an edge cutting of the polyhedron, and the resulting simple polygon is called an edge unfolding or net.
The authors investigate the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. Two examples of polyhedra are given, one with 24 convex faces and one with 36 triangular faces that cannot be unfolded by cutting along edges. This proves, in particular, that the edge-unfolding conjecture does not generalize to topologically convex polyhedra. It is shown that cuts across faces can unfold some convex-faced polyhedra that cannot be unfolded with cuts only along edges. This is the first demonstration that general cuts are more powerful than edge cuts.
The problem of constructing a polyhedron that cannot be unfolded even using general cuts is also considered. Finding a “closed” ununfoldable polyhedron is an intriguing open problem. As a step towards this goal, the authors present an “open” polyhedron with triangular faces that cannot be unfoldable no matter how it is cut.