Programming shape-changing materials: folding, cutting, inflating

Cartographers have early realized that it is impossible to draw a flat map of the Earth without deforming continents. More generally, mapping a doubly curved surfaces such as a dome or a saddle shape into a flat surface, necessarily involves in-plane distortions. This geometrical constraint was rationalized by Gauss in his seminal Theorema Egregium. Can we invert the problem and obtain a 3D shape by changing the local distances in an initially flat plate? Nature is full of complex shapes such a petals or leaves obtained by differential growth. What strategies may engineers deploy to modify distances in a plane? We will review different distortion techniques ranging from folding or cutting paper (origami or kirigami), inflating fabrics or patches of elastomer with embedded channels. Can we program the shapes of the deployed structures?