Unfolding a polyhedron can be thought of as applying a series of cuts to its surface so that it can be unfolded into a single piece that can be laid flat without overlap. Often what is desired is an edge unfolding in which the cuts are restricted to the polyhedron's edges. People have been interested in finding edge unfoldings since at least 1525 when painter and printmaker Albrecht Dürer published a book containing edge unfoldings for many convex polyhedra. It is implicit in Dürer's work (and much subsequent work) that every convex polyhedron has an edge unfolding, but in fact this is still an open question today. An answer to this question would be interesting not only for pure intellectual reasons, but also because this and related questions arise in manufacturing processes that construct three-dimensional objects by bending (folding) sheet metal. Interest in this topic is currently growing in the computational geometry research community, and there is a rich set of open problems to work on.
In this talk I will introduce polyhedra unfolding and discuss both edge and general unfoldings, unfoldings of non-convex polyhdedra, unfolding algorithms, and some open problems suitable for undergraduates to work on.