Skip to main content
SpringerLink
Log in

Continuous flattening of the 2-skeletons in regular simplexes and cross-polytopes

  • Published:
Journal of Geometry Aims and scope Submit manuscript

Abstract

We previously showed one can continuously flatten the surface of a regular tetrahedron onto any of its faces without stretching and cutting. This is accomplished by moving creases to change the shapes of some faces successively, following Sabitov’s volume preserving theorem. We extend this result to higher dimensional regular simplexes and cross-polytopes by considering the 2-dimensional skeleton of a polytope corresponding to the surface of a three dimensional polyhedron.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  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 30th Annual Symposium on Computational Geometry (SoCG), pp. 396–405. (2014)

  2. Connelly, R., Sabitov, I., Walz, A.: The bellows conjecture. Beiträge Algebra Geom. 38, 1–10 (1997)

    MathSciNet  MATH  Google Scholar 

  3. Demaine, E.D., Demaine, M.D., Lubiw, A.: Flattening polyhedral. Unpublished manuscript, (2001)

  4. Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  5. Itoh, J., Nara, C.: Continuous flattening of Platonic polyhedra, In: Proceedings of Computational Geometry, Graphs, and Applications (CGGA 2010), LNCS, vol. 7033, pp. 108–121. Springer, Berlin (2011)

    Google Scholar 

  6. Itoh, J., Nara, C.: Continuous flattening of the 2-dimensional skeleton of the square faces in a hypercube. Submitted.

  7. Itoh, J., Nara, C.: Continuous flattening of truncated tetrahedra. J. Geom. 107(1), 61–75 (2016)

    Article  MathSciNet  Google Scholar 

  8. Itoh, J., Nara, C., Vîlcu, C.: Continuous flattening of convex polyhedra. In: Revised Papers, 16th Spanish Meeting on Computational Geometry (EGC 2011), LNCS, vol. 7579, pp. 85–97. Springer, Berlin (2012)

    Google Scholar 

  9. Nara, C.: Continuous flattening of some pyramids. Elem. Math. 69, 45–56 (2014)

    Article  MathSciNet  Google Scholar 

  10. Sabitov, I.: The volume of polyhedron as a function of its metric. Fundam. Prikl. Mat. 2, 1235–1246 (1996)

    MathSciNet  MATH  Google Scholar 

  11. Sabitov, I.: The volume as a metric invariant of polyhedra. Discret. Comput. Geom. 20, 405–425 (1998)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are indebted to Joseph O’Rourke for his careful reading of a preliminary version of this paper and his valuable suggestions, particularly improving the proof of Lemma 3.1 and Fig. 6a.

Author information

Authors and Affiliations

  1. School of Education, Sugiyama Jyogakuen University, Chikusa-ku, Nagoya, Japan

    Jin-ichi Itoh

  2. Organization for the Strategic Coordination of Research and Intellectual Properties, Meiji University, Nakano, Tokyo,  164-8525, Japan

    Chie Nara

Authors
  1. Jin-ichi Itoh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chie Nara
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chie Nara.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Jin-ichi Itoh was partially supported by Grant-in-Aid for Scientific Research (B) (No. 15KT0020), Japan Society for the Promotion of Science.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Itoh, Ji., Nara, C. Continuous flattening of the 2-skeletons in regular simplexes and cross-polytopes. J. Geom. 110, 47 (2019). https://doi.org/10.1007/s00022-019-0504-0

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00022-019-0504-0

Keywords

Mathematics Subject Classification

Search

Navigation