Internal Continuous Flattening of Polyhedra

  • Kazuki MatsubaraEmail author
  • Chie Nara
Original paper


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).


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



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.


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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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