Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami

We show a remarkable fact about folding paper: From a single square of paper, one can fold it into a flat origami that takes the (scaled) shape of any connected polygonal region, even if it has holes. This resolves a long-standing open problem in origami design. Our proof is constructive, utilizing tools of computational geometry, resulting in efficient algorithms for achieving the target silhouette. We show further that if the paper has a different color on each side, we can form any connected polygonal pattern of two colors. Our results apply also to polyhedral surfaces, showing that any polyhedron can be `wrapped' by folding a strip of paper around it. We give three methods for solving these problems: the first uses a thin strip whose area is arbitrarily close to optimal; the second allows wider strips to be used; and the third varies the strip width to make a folding that optimizes the number or length of visible `seams.'