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Internal Continuous Flattening of Polyhedra

  • Kazuki MatsubaraEmail author
  • Chie Nara
Original paper
  • 9 Downloads

Abstract

There are several ways to continuously flatten polyhedra. In this paper, we focus on continuous flattening with the special property that every intermediate state is contained in the interior of the original polyhedron. The orderly squashing method and the \(\alpha \)-belt method given in the literature are useful for internal continuous flattening of polyhedra. We investigate these methods, and then provide internal continuous flattening motions for several types of convex polyhedra. Moreover, we give new methods of continuous flattening of prisms whose moving creases exist only on the top and bottom faces. Finally, we show that an internal continuous flattening motion can be obtained for any prism (not necessarily convex).

Keywords

Polyhedron Internal continuous flattening Orderly squashing \(\alpha \)-Belt Moving crease Prism 

Notes

Acknowledgements

The second author is supported by Grant-in-Aid for Scientific Research (C)(16K05258). The authors would like to thank referees for constructive comments. They made the paper much more accurate and readable.

References

  1. 1.
    Abel, Z., Demaine, E.D., Demaine, M.L., Itoh, J., Lubiw, A., Nara, C., O’Rourke, J.: Continuously flattening polyhedra using straight skeletons. In: Proc. 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)MathSciNetCrossRefGoogle 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., Itoh, J., Nara, C.: Continuous flattening of orthogonal polyhedra. In: Proc. JCDCGG 2015, LNCS, vol. 9943, pp. 85–93. Springer (2016)Google Scholar
  5. 5.
    Demaine, E.D., Demaine, M.L., Lubiw, A.: Flattening polyhedra, unpublished manuscript (2001)Google Scholar
  6. 6.
    Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, Cambridge (2007)CrossRefGoogle Scholar
  7. 7.
    Itoh, J., Nara, C.: Continuous flattening of Platonic polyhedra. In: Proc. Computational Geometry, Graphs and Applications (CGGA 2010), LNCS, vol. 7033, pp. 108–121. Springer (2011)Google Scholar
  8. 8.
    Itoh, J., Nara, C., Vîlcu, C.: Continuous flattening of convex polyhedra. In: Proc. 16th Spanish Meeting on Computational Geometry (EGC 2011), LNCS, vol. 7579, pp. 85–97. Springer (2012)Google Scholar
  9. 9.
    Matsubara, K., Nara, C.: Continuous flattening of \(\alpha \)-trapezoidal polyhedra. J. Inf. Process. 25, 554–558 (2017)Google Scholar
  10. 10.
    Meisters, G.H.: Polygons have ears. Am. Math. Mon. 82, 648–651 (1975)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Nara, C.: Continuous flattening of some pyramids. Elem. Math. 69(2), 45–56 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Chuo Gakuin UniversityChibaJapan
  2. 2.Meiji Institute for Advanced Study of Mathematical SciencesTokyoJapan

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