In this talk we explore the crushing of an thin elastic membrane into a small sphere. Such crushing occurs when macroscopic sheets are crumpled or when microscopic membranes are distorted by osmotic or hydrodynamic forces. Crushing produces singular points of high curvature. We study the interaction between these singular vertices by energy-balance arguments and by numerical simulations. We find that two vertices are connected by a characteristic ridge, whose width and radius of curvature scale as the two-thirds power of the distance between the vertices. The associated deformation energy scales as the one-third power of this distance. Thus large, thin sheets have their deformation energy concentrated in an arbitrarily small fraction of the sheet. Still, the deformation is weak enough that the sheet may be accurately described by linear elasticity. With these features in mind we discuss the energy of confinement for a crushed sheet. The generalization of this crushing to higher spatial dimensions yields a surprise: the crushing can break symmetry to produce preferred directions in the membrane.