Paper by Erik D. Demaine

Erik D. Demaine, Martin L. Demaine, John Iacono, and Stefan Langerman, “Wrapping Spheres with Flat Paper”, Computational Geometry: Theory and Applications, volume 42, number 8, 2009, pages 748–757. Special issue of selected papers from the 20th European Workshop on Computational Geometry, 2007.

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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The paper is 14 pages.

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Related papers:
SphereWrapping_EuroCG2007 (Wrapping the Mozartkugel)

See also other papers by Erik Demaine.
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Last updated October 16, 2017 by Erik Demaine.