Pneumatic cells toward absolute Gaussian morphing

On a flat map of the Earth, continents are inevitably distorted. Reciprocally, curving a plate simultaneously in two directions requires a modification of in-plane distances, as Gauss stated in his seminal theorem. Although emerging architectured materials with programmed in-plane distortions are capable of such shape morphing, an additional control of local bending is required to precisely set the final shape of the resulting three-dimensional surface. Inspired by bulliform cells in leaves of monocotyledon plants, we show how the internal structure of flat panels can be designed to program bending and in-plane distortions simultaneously when pressurized, leading to a targeted shell shape. These surfaces with controlled stiffness and fast actuation are manufactured using consumer-grade materials and open a route to large-scale shape-morphing robotics applications.