Soft living organisms have perfected the use of local deformations for dynamic global shape control. Recapitulating this in responsive synthetic materials involves programming a differential swelling field into the material. This field determines the material’s elastic state upon actuation and can be mathematically encoded as a reference Riemannian metric tensor. In this framework, the elastic problem reduces to finding distortion-minimizing embeddings of the reference metric into physical space. This geometric perspective reveals a rich mathematical structure connecting differential geometry, calculus of variations, and asymptotic analysis to materials science. I present two complementary results on programmable thin sheets. First, I demonstrate that sharp swelling gradients, coupled with spatial variations in bending rigidity, produce spontaneous wrinkling: undulations with emergent small characteristic length scales that arise without external constraints. These wrinkles are spatially localized and follow novel scaling laws, reproducing patterns observed at the edges of leaves and petals. Second, I present a fabrication scheme using 4D printing (3D printing with responsive materials) that achieves full geometric control by simultaneously programming both Gaussian and mean curvatures of the reference geometry. This enables control over not only the shape but also the mechanical response of the material. Together, these results demonstrate new pathways for designing shape-morphing materials through geometric programming.

Refreshments will be served preceding the talk, beginning at 2:45.