Using Growth and Folding to Shape Elastic Sheets

Despite their everyday familiarity, thin sheets (paper, plastic, fabric, etc.) display remarkable and complex behaviors that still challenge theoretical description. The intricate coupling between the geometry of surfaces and the elasticity of a thin sheet necessarily leads to the formation of singularities, nonlinear elasticity, and geometric frustration. Nevertheless, multicellular organisms - like you - develop their three dimensional structures in part by exploiting these elastic phenomena. These considerations have led to new theoretical and experimental tools to shape elastic sheets into prescribed 3D shapes using the principles of non-Euclidean geometry. I will describe our attempts to design sheets that fold controllably into 3D structures and some related problems in the mechanics of origami, where 3D structure is developed by folding a piece of paper. These techniques open up new avenues in "experimental mathematics", allowing us to explore geometry experimentally.