Wrapping spheres with flat paper

Erik D. Demaine, Martin L. Demaine, John Iacono, Stefan Langerman

    Research output: Contribution to journalArticle

    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.

    Original languageEnglish (US)
    Pages (from-to)748-757
    Number of pages10
    JournalComputational Geometry: Theory and Applications
    Volume42
    Issue number8
    DOIs
    StatePublished - Oct 2009

    Keywords

    • Contractive mapping
    • Folding
    • Mozartkugel
    • Sphere

    ASJC Scopus subject areas

    • Computer Science Applications
    • Geometry and Topology
    • Control and Optimization
    • Computational Theory and Mathematics
    • Computational Mathematics

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  • Cite this

    Demaine, E. D., Demaine, M. L., Iacono, J., & Langerman, S. (2009). Wrapping spheres with flat paper. Computational Geometry: Theory and Applications, 42(8), 748-757. https://doi.org/10.1016/j.comgeo.2008.10.006