INTRODUCTION

We return to Open Problem 21.1 (p. 300): can every convex polyhedron be cut along its edges and unfolded flat into the plane to a single nonoverlapping simple polygon, a net for the polyhedron? This chapter explores the relatively meager evidence for and against a positive answer to this question, as well as several more developed, tangentially related topics.

Applications in Manufacturing

Although this problem is pursued primarily for its mathematical intrigue, it is not solely of academic interest: manufacturing parts from sheet metal (cf. Section 1.2.2, p. 13) leads directly to unfolding issues. A 3D part is approximated as a polyhedron, its surface is mapped to a collection of 2D flat patterns, each is cut from a sheet of metal and folded by a bending machine (Kim et al. 1998), and the resulting pieces assembled to form the final part. Clearly it is essential that the unfolding be nonoverlapping and great efficiency is gained if it is a single piece. The author of a Ph.D. thesis in this area laments that “Unfortunately, there is no theorem or efficient algorithm that can tell if a given 3D shape is unfoldable [without overlap] or not” (Wang 1997, p. 81). In general, those in manufacturing are most keenly interested in unfolding nonconvex polyhedra and, given the paucity of theoretical results to guide them, have relied on heuristic methods. One of the more impressive commercial products is TouchCAD by Lundströom Design, which has been used, for example, to design a one-piece vinyl cover for mobile phones (see Figure 22.1).