Proceedings of the Annual Symposium on Computational Geometry
We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a at folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The attening respects the straight-skeleton gluing, meaning that points of the polyhedron touched by a common ball inside the polyhedron come into contact in the at folding, which answers an open question in the book Geometric Folding Algorithms. The primary creases in our folding process can be found in quadratic time, though necessarily, creases must roll continuously, and we show that the full crease pattern can be exponential in size. We show that our method solves the fold-and-cut problem for convex polyhedra in any dimension. As an additional application, we show how a limiting form of our algorithm gives a general design technique for at origami tessellations, for any spiderweb (planar graph with all-positive equilibrium stress).
Attening, Fold-and-cut, Folding, Medial axis, Origami, Straight skeleton, Tessellations
© the authors
Abel, Zachary; Demaine, Erik D.; Demaine, Martin L.; Itoh, Jin Ichi; Lubiw, Anna; Nara, Chie; and O'Rourke, Joseph, "Continuously Flattening Polyhedra Using Straight Skeletons" (2014). Computer Science: Faculty Publications, Smith College, Northampton, MA.