Folding a Better Checkerboard

  • Erik D. Demaine
  • Martin L. Demaine
  • Goran Konjevod
  • Robert J. Lang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5878)

Abstract

Folding an n ×n checkerboard pattern from a square of paper that is white on one side and black on the other has been thought for several years to require a paper square of semiperimeter n2. Indeed, within a restricted class of foldings that match all previous origami models of this flavor, one can prove a lower bound of n2 (though a matching upper bound was not known). We show how to break through this barrier and fold an n ×n checkerboard from a paper square of semiperimeter \({1 \over 2} n^2 + O(n)\). In particular, our construction strictly beats semiperimeter n2 for (even) n > 16, and for n = 8, we improve on the best seamless folding.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Casey, S.: Chess board. In: Proceedings of the West Coast Origami Guild, vol. 19, pp. 3–12 (August 1989)Google Scholar
  2. 2.
    Chen, S.Y.: Checker board. In: Proceedings of the Annual OUSA Convention, pp. 72–75 (2001), http://www.origami-usa.org/files/CheckerBoard.PDF
  3. 3.
    Demaine, E.D., Demaine, M.L., Mitchell, J.S.B.: Folding flat silhouettes and wrapping polyhedral packages: New results in computational origami. Computational Geometry: Theory and Applications 16(1), 3–21 (2000)MATHMathSciNetGoogle Scholar
  4. 4.
    Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, Cambridge (July 2007)MATHGoogle Scholar
  5. 5.
    Dureisseix, D.: Chessboard. British Origami 201, 20–24 (2000)Google Scholar
  6. 6.
    Fujimoto, S.: Crowding butterflies. In: BOS Autumn Convention, p. 27 (1989)Google Scholar
  7. 7.
    Guy, M., Venables, D.: The Chess Sets of Martin Wall, Max Hulme and Neal Elias. British Origami Society Booklet 7 (1979)Google Scholar
  8. 8.
    Kirschenbaum, M.: Chess board. The Paper 61, 24–30 (1998), http://dev.origami.com/images_pdf/chessboard.pdf Google Scholar
  9. 9.
    Lang, R.J.: Origami Design Secrets: Mathematical Methods for an Ancient Art. AK Peters, Wellesley (2003)MATHGoogle Scholar
  10. 10.
    Montroll, J.: Origami Inside-Out. Dover, Newyork (1993)Google Scholar
  11. 11.
    Yoshizawa, A.: Origami Museum I. Kamakura Shobo Publishing Co., Tokyo (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • Martin L. Demaine
    • 1
  • Goran Konjevod
    • 2
  • Robert J. Lang
    • 3
  1. 1.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
  2. 2.Department of Computer Science and EngineeringArizona State UniversityTempeUSA
  3. 3.No Affiliations 

Personalised recommendations