Overlapping Unfoldings of Cones and Convex Polyhedra

2026-07-10

AI Summary

This work revisits the classical Dürer unfolding problem—traditionally concerned with avoiding overlaps—and instead investigates how to construct unfoldings of convex polyhedra that realize a prescribed overlap thickness. We present the first systematic framework combining geometric and combinatorial techniques, encompassing both edge cuts and non-edge cuts. Our main contributions are twofold: we prove that for any given overlap thickness, there exists a convex polyhedron whose unfolding achieves this thickness via non-edge cuts; furthermore, for every positive integer thickness, we demonstrate the existence of a convex polyhedron that attains the specified overlap thickness using only edge-unfolding cuts, thereby transcending the conventional constraint of overlap-free unfoldings.

Abstract

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.

Problem

• overlapping unfoldings
• convex polyhedra
• Dürer's problem
• edge unfoldings
• unfolding thickness

Innovation

• overlapping unfolding
• convex polyhedra
• Dürer's problem
• edge unfolding
• unfolding thickness