Wrapping spheres with flat paper |
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Authors: | Erik D Demaine Martin L Demaine John Iacono Stefan Langerman |
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Institution: | 1. MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, MA 02139, USA;2. Department of Computer Science and Engineering, Polytechnic Institute of New York University, 5 MetroTech Center, Brooklyn, NY 11201, USA;3. Maître de recherches, Département d''informatique, Université Libre de Bruxelles, ULB CP212, 1050 Brussels, Belgium |
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Abstract: | We study wrappings of smooth (convex) surfaces by a flat piece of paper or foil. Such wrappings differ from standard mathematical origami because they require infinitely many infinitesimally small folds (“crumpling”) in order to transform the flat sheet into a surface of nonzero curvature. Our goal is to find shapes that wrap a given surface, have small area and small perimeter (for efficient material usage), and tile the plane (for efficient mass production). Our results focus on the case of wrapping a sphere. We characterize the smallest square that wraps the unit sphere, show that a 0.1% smaller equilateral triangle suffices, and find a 20% smaller shape contained in the equilateral triangle that still tiles the plane and has small perimeter. |
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