Significance What makes an object successful at thermal folding? Protein scientists study how sequence affects the pathways by which chained amino acids fold and the structures into which they fold. Here we investigate the inverse problem: Starting with a 3D object as a polyhedron we ask, which ones, among the many choices of 2D unfoldings, are able to fold most consistently? We find that these “nets” follow a universal balance between entropy loss and potential energy gain, allowing us to explain why some of their geometrical attributes (such as compactness) represent a good predictor for the folding propensity of a given shape. Our results can be used to guide the stochastic folding of nanoscale objects into drug-delivery devices and thermally folded robots.