Paper by Erik D. 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.

BibTeX

@Article{SphereWrapping_CGTA,
  AUTHOR        = {Erik D. Demaine and Martin L. Demaine and John Iacono and
                   Stefan Langerman},
  TITLE         = {Wrapping Spheres with Flat Paper},
  JOURNAL       = {Computational Geometry: Theory and Applications},
  journalurl    = {https://www.sciencedirect.com/journal/computational-geometry},
  VOLUME        = 42,
  NUMBER        = 8,
  YEAR          = 2009,
  PAGES         = {748--757},
  NOTE          = {Special issue of selected papers from the
                   20th European Workshop on Computational Geometry, 2007.},
  doi           = {https://dx.doi.org/10.1016/J.COMGEO.2008.10.006},
  dblp          = {https://dblp.org/rec/journals/comgeo/DemaineDIL09},
  comments      = {This paper is also available from <A HREF="http://dx.doi.org/10.1016/j.comgeo.2008.10.006">ScienceDirect</A>.},
  papers        = {SphereWrapping_EuroCG2007},
  length        = {14 pages},
}

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:

The paper is available in gzipped PostScript (5289k) and PDF (508k).

See information on file formats.

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Related papers:

SphereWrapping_EuroCG2007 (Wrapping the Mozartkugel)