Making Polygons by Simple Folds and One Straight Cut

  • Erik D. Demaine
  • Martin L. Demaine
  • Andrea Hawksley
  • Hiro Ito
  • Po-Ru Loh
  • Shelly Manber
  • Omari Stephens
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7033)

Abstract

We give an efficient algorithmic characterization of simple polygons whose edges can be aligned onto a common line, with nothing else on that line, by a sequence of all-layers simple folds. In particular, such alignments enable the cutting out of the polygon and its complement with one complete straight cut. We also show that these makeable polygons include all convex polygons possessing a line of symmetry.

Keywords

Convex Hull Convex Polygon Steiner Point Simple Polygon Polygonal Line 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Arkin, E.M., Bender, M.A., Demaine, E.D., Demaine, M.L., Mitchell, J.S.B., Sethia, S., Skiena, S.S.: When can you fold a map? Computational Geometry: Theory and Applications 29(1), 23–46 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bern, M., Demaine, E., Eppstein, D., Hayes, B.: A disk-packing algorithm for an origami magic trick. In: Origami3: Proceedings of the 3rd International Meeting of Origami Science, Math, and Education, Monterey, California, pp. 17–28 (March 2001); Improvement of version appearing in Proceedings of the International Conference on Fun with Algorithms, Isola d’Elba, Italy, pp. 32–42 (June 1998) Google Scholar
  3. 3.
    Demaine, E.D., Demaine, M.L., Lubiw, A.: Folding and Cutting Paper. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 104–118. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Demaine, E.D., Devadoss, S.L., Mitchell, J.S.B., O’Rourke, J.: Continuous foldability of polygonal paper. In: Proceedings of the 16th Canadian Conference on Computational Geometry, Montréal, Canada, pp. 64–67 (August 2004)Google Scholar
  6. 6.
    Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press (July 2007)Google Scholar
  7. 7.
    Melkman, A.A.: On-line construction of the convex hull of a simple polyline. Information Processing Letters 25(1), 11–12 (1987)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Wolter, J.D., Woo, T.C., Volz, R.A.: Optimal algorithms for symmetry detection in two and three dimensions. The Visual Computer 1(1), 37–48 (1985)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • Martin L. Demaine
    • 1
  • Andrea Hawksley
    • 1
  • Hiro Ito
    • 2
  • Po-Ru Loh
    • 1
  • Shelly Manber
    • 1
  • Omari Stephens
    • 1
  1. 1.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
  2. 2.School of InformaticsKyoto UniversityKyotoJapan

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