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

    Fingerprint

    Equilateral triangle
    Perimeter
    Tile
    Convex Surface
    Unit Sphere
    Fold
    Curvature
    Transform
    Metal foil
    Standards

    Keywords

    • Contractive mapping
    • Folding
    • Mozartkugel
    • Sphere

    ASJC Scopus subject areas

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

    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

    Wrapping spheres with flat paper. / Demaine, Erik D.; Demaine, Martin L.; Iacono, John; Langerman, Stefan.

    In: Computational Geometry: Theory and Applications, Vol. 42, No. 8, 10.2009, p. 748-757.

    Research output: Contribution to journalArticle

    Demaine, ED, Demaine, ML, Iacono, J & Langerman, S 2009, 'Wrapping spheres with flat paper', Computational Geometry: Theory and Applications, vol. 42, no. 8, pp. 748-757. https://doi.org/10.1016/j.comgeo.2008.10.006
    Demaine, Erik D. ; Demaine, Martin L. ; Iacono, John ; Langerman, Stefan. / Wrapping spheres with flat paper. In: Computational Geometry: Theory and Applications. 2009 ; Vol. 42, No. 8. pp. 748-757.
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