Unfolding Prismatoids as Convex Patches: Counterexamples and Positive Results

We address the unsolved problem of unfolding prismatoids in a new context, viewing a "topless prismatoid" as a convex patch---a polyhedral subset of the surface of a convex polyhedron homeomorphic to a disk. We show that several natural strategies for unfolding a prismatoid can fail, but obtain a positive result for "petal unfolding" topless prismatoids. We also show that the natural extension to a convex patch consisting of a face of a polyhedron and all its incident faces, does not always have a nonoverlapping petal unfolding. However, we obtain a positive result by excluding the problematical patches. This then leads a positive result for restricted prismatoids. Finally, we suggest suggest studying the unfolding of convex patches in general, and offer some possible lines of investigation.