Distortion Controlled Differential Growth

Non-Euclidean plates, thin elastic sheets that grow or shrink inhomogeneously, can be thought of surfaces which are unable to completely eliminate in-plane stress. This geometrical frustration forces them to adopt interesting rest configurations as a way of reducing the elastic energy of the system. In this work, we study conformal flattening as a tool for prescribing desirable patterns of nonuniform growth on elastic sheets as a way of making them buckle into a given target shape. In particular, we explore how to design a planar shape that requires the least amount of inhomogeneous growth to buckle into a sphere. We tune the ratio of maximal to minimal area distortion required by modifying where the seams of the final spherical shape will be. We then discuss the optimality of the cuts and make some conjectures about the best possible cuts on arbitrary surfaces.