Overcurvature describes the buckling and folding of rings from curved origami to foldable tents

Daily-life foldable items, such as popup tents, the curved origami sculptures exhibited in the Museum of Modern Art of New York, overstrained bicycle wheels, released bilayered microrings and strained cyclic macromolecules, are made of rings buckled or folded in tridimensional saddle shapes. Surprisingly, despite their popularity and their technological and artistic importance, the design of such rings remains essentially empirical. Here we study experimentally the tridimensional buckling of rings on folded paper rings, lithographically processed foldable microrings, human-size wood sculptures or closed arcs of Slinky springs. The general shape adopted by these rings can be described by a single continuous parameter, the overcurvature. An analytical model based on the minimization of the energy of overcurved rings reproduces quantitatively their shape and buckling behaviour. The model also provides guidelines on how to efficiently fold rings for the design of space-saving objects.