Paper by Martin L. Demaine

Reference:

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.

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.

Comments:

This paper is also available from ScienceDirect.

Length:

The paper is 14 pages.

Availability:

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