Paper by Martin L. Demaine

Reference:

Zachary Abel, Erik D. Demaine, Martin L. Demaine, Jin-Ichi Itoh, Anna Lubiw, Chie Nara, and Joseph O'Rourke, “Continuously Flattening Polyhedra Using Straight Skeletons”, in Proceedings of the 30th Annual Symposium on Computational Geometry (SoCG 2014), Kyoto, Japan, June 8–11, 2014, pages 396–405.

Abstract:

We prove that a surprisingly simple algorithm folds the surface of every convex polyhedron, in any dimension, into a flat folding by a continuous motion, while preserving intrinsic distances and avoiding crossings. The flattening respects the straight-skeleton gluing, meaning that points of the polyhedron touched by a common ball inside the polyhedron come into contact in the flat 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 flat origami tessellations, for any spiderweb (planar graph with all-positive equilibrium stress).

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