Overlapping Unfoldings of Cones and Convex Polyhedra
For mathematicians and computational geometers, this provides a new perspective on unfolding polyhedra, though it is an incremental extension of existing problems.
The paper inverts Dürer's problem to find unfoldings of convex polyhedra that overlap to any given thickness, proving that both general and edge unfoldings can achieve arbitrary overlap thickness.
Research on Dürer's problem focuses on edge unfoldings of convex polyhedra that avoid overlap. We invert the goal and find unfoldings that overlap at some point to any given thickness t. We have two main results. The first is that, if we allow unfolding cuts that do not follow polyhedron edges, then there is a convex polyhedron that can unfold with overlap of any given thickness. The second result is that for any given thickness, there is a convex polyhedron with an edge unfolding that overlaps to that thickness.