Thin elastic membranes form complex wrinkle patterns when put on substrates of different shapes. Such patterns continue to receive attention across science and engineering. This is due, in part, to the promise of lithography-free micropatterning, but also to the observation that similar patterns arise in biological systems from growth. The challenge is to explain the patterns in any given setup, even when they fail to be robust. Building on the theoretical foundation of [Tobasco, to appear in Arch. Ration. Mech. Anal., arXiv:1906.02153], we derive a complete and simple rule set for wrinkles in the model system of a curved shell on a liquid bath. Our rules apply to shells whose initial Gaussian curvatures are of one sign, such as cutouts of saddles and spheres. They predict the surprising coexistence of orderly wrinkles alongside disordered regions where the response appears stochastic, which we confirm in experiment and simulation. They also unveil the role of the shells medial axis, a distinguished locus of points that we show is a basic driver in pattern selection. Finally, they explain how the sign of the shells initial curvature dictates the presence or lack of disorder. Armed with our simple rules, and the methodology underlying them, one can anticipate the creation of designer wrinkle patterns.
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