What causes pinecones to open and close? How do lily petals change shape as they bloom? General morphing of such surfaces into complex 3D shapes involves modifying in-plane distances (i.e., metric distortion). Cartographers have long understood this, as continents are inevitably distorted on flat maps of Earth. Carl Gauss further formalized this geometric constraint in his seminal theorem, referred to here as Gaussian morphing. In natural structures, metric changes result from differential growth, while engineered systems use non-homogeneous transformations like hydrogel swelling, liquid crystal elastomer relaxation, or origami tessellation. However, simply imposing metrics is not enough to set the final shape precisely; additional control of local bending is necessary to select between different isometric shapes. In this talk, I will present four strategies for shape morphing: (1) inflatable zigzag meso-structures with biaxial active stretches; (2) pneumatic Gaussian cells with non-symmetric cross-section channels; (3) self-folding structures along curved creases via a classical bilayer effect; (4) fish fin inspired composite beam combining high flexural stiffness and high morphing efficiency. Central to all these strategies is the interplay between geometry and elasticity, offering avenues to manipulate both metric and/or bending for on-demand shape transformations.

Work performed in collaboration with José Bico and Benoit Roman (PMMH, ESPCI, CNRS, UPC, PSU, PSL)