An Additive Algorithm For Origami Design

Folding patterns arise in nature in systems including insect wings, leaves, and guts (1-4) and have a long history in decorative, ceremonial, and pedagogical traditions of origami around the world. More recently, they have begun to draw the attention of mathematicians fascinated by the patterns and limits of folding (5-9) and engineers and scientists inspired by their technological promise (10-18). The simplest origami is a single vertex with four folds, a kindInspired by the allure of additive fabrication, we pose the problem of origami design from a different perspective: How can we grow a folded surface in three dimensions from a seed so that it is guaranteed to be isometric to the plane? We solve this problem in two steps: by first identifying the geometric conditions for the compatible completion of two separate folds into a single developable fourfold vertex, and then showing how this foundation allows us to grow a geometrically compatible front at the boundary of a given folded seed. This yields a complete marching, or additive, algorithm for the inverse design of the complete space of developable quad origami patterns that can be folded from flat sheets. We illustrate the flexibility of our approach by growing ordered, disordered, straight, and curved-folded origami and fitting surfaces of given curvature with folded approximants. Overall, our simple shift in perspective from a global search to a local rule has the potential to transform origami-based metastructure design.