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Continuous Flattening of Orthogonal Polyhedra

  • Erik D. Demaine
  • Martin L. Demaine
  • Jin-ichi Itoh
  • Chie NaraEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9943)

Abstract

Can we flatten the surface of any 3-dimensional polyhedron P without cutting or stretching? Such continuous flat folding motions are known when P is convex, but the question remains open for nonconvex polyhedra. In this paper, we give a continuous flat folding motion when the polyhedron P is an orthogonal polyhedron, i.e., when every face is orthogonal to a coordinate axis (x, y, or z). More generally, we demonstrate a continuous flat folding motion for any polyhedron whose faces are orthogonal to the z axis or the xy plane.

Keywords

Folding Continuous flattening Orthogonal polyhedra 

References

  1. 1.
    Abel, Z., Demaine, E.D., Demaine, M.L., Itoh, J.-I., Lubiw, A., Nara, C., O’Rourke, J.: Continuously flattening polyhedra using straight skeletons. In: Proceedings of the 30th Annual Symposium on Computational Geometry (SoCG), pp. 396–405 (2014)Google Scholar
  2. 2.
    Bern, M., Hayes, B.: Origami embedding of piecewise-linear two-manifolds. Algorithmica 59(1), 3–15 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Connelly, R., Sabitov, I., Walz, A.: The bellows conjecture. Beiträge Algebra Geom. 38, 1–10 (1997)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Demaine, E.D., Demaine, M.L., Lubiw, A.: Flattening polyhedra (2001). Unpublished manuscriptGoogle Scholar
  5. 5.
    Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  6. 6.
    Itoh, J., Nara, C.: Continuous flattening of platonic polyhedra. In: Akiyama, J., Bo, J., Kano, M., Tan, X. (eds.) CGGA 2010. LNCS, vol. 7033, pp. 108–121. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-24983-9_11 CrossRefGoogle Scholar
  7. 7.
    Itoh, J., Nara, C., Vîlcu, C.: Continuous flattening of convex polyhedra. In: Márquez, A., Ramos, P., Urrutia, J. (eds.) EGC 2011. LNCS, vol. 7579, pp. 85–97. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-34191-5_8 CrossRefGoogle Scholar
  8. 8.
    Nara, C.: Continuous flattening of some pyramids. Elem. Math. 69(2), 45–56 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • Martin L. Demaine
    • 1
  • Jin-ichi Itoh
    • 2
  • Chie Nara
    • 3
    Email author
  1. 1.MIT Computer Science and Artificial Intelligence LaboratoryCambridgeUSA
  2. 2.Faculty of EducationKumamoto UniversityKumamotoJapan
  3. 3.Meiji Institute for Advanced Study of Mathematical SciencesMeiji UniversityTokyoJapan

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