Skip to main content
Log in

On the model flexibility of Siamese dipyramids

  • Published:
Journal of Geometry Aims and scope Submit manuscript

Abstract

Polyhedra called Siamese dipyramids are known to be non-flexible, however their physical models behave like physical models of flexible polyhedra. We discuss a simple mathematical method for explaining the model flexibility of the Siamese dipyramids.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Connelly, R.: A counterexample to the rigidity conjecture for polyhedra. Publications Mathematiques de l’IHES 47, 333–338 (1997)

    Article  Google Scholar 

  2. Connelly, R.: Rigidity. Handbook of Convex Geometry, vol. A, pp. 223–271. North-Holland, Amsterdam (1993)

    Book  Google Scholar 

  3. Connelly, R.: The rigidity of certain cabled frameworks and the second-order rigidity of arbitrarily triangulated convex surface. Adv. Math. 31, 212–299 (1980)

    MATH  Google Scholar 

  4. Cromwell, P.: Polyhedra. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  5. Fuchs, D., Tabachnikov, S.: Mathematical Omnibus: Thirty Lectures on Classic Mathematics. American Mathematical Society, Providence (2007)

    Book  Google Scholar 

  6. Gorkavyy, V., Kalinin, D.: On model flexibility of the Jessen orthogonal icosahedron. Contrib. Algebra Geom. 57, 607–622 (2016)

    Article  MathSciNet  Google Scholar 

  7. Gluck, H.: Almost all simply connected surfaces are rigid. Geometric Topology. Lecture Notes in Mathematics, vol. 438, pp. 225–239. Springer, Berlin/New York (1975)

  8. Goldberg, M.: Unstable polyhedral structures. Math. Mag. 51, 165–170 (1978)

    Article  MathSciNet  Google Scholar 

  9. Milka, A.D.: Nonrigid star-like bipiramids of A.D. Alexandrov and S.M. Vladimirova. Sib. Adv. Math. 12(2), 56–72 (2002)

    MathSciNet  MATH  Google Scholar 

  10. Milka, A.D.: Bendings of surfaces, bifurcations of dynamical systems, and the stability of shells. Chebyshevskii Sbornik 7(2), 109–144 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Milka, A.D.: Linear bendings of star-like bipyramids. C.R. Mecanique 331, 805–810 (2003)

    Article  Google Scholar 

  12. Milka, A.D.: Linear bendings of star-like bipyramids. Eur. J. Comb. 31, 1050–1064 (2010)

    Article  MathSciNet  Google Scholar 

  13. Pogorelov, A.V.: Bendings of Surfaces and Stability of Shells. Translations of Mathematical Monographs, vol. 72. American Mathematical Society, Providence (1988)

    Book  Google Scholar 

  14. Wunderlich, W.: Snapping and shaky antiprisms. Math. Mag. 52, 235–236 (1979)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to V. Gorkavyy.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gorkavyy, V., Fesenko, I. On the model flexibility of Siamese dipyramids. J. Geom. 110, 7 (2019). https://doi.org/10.1007/s00022-018-0462-y

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1007/s00022-018-0462-y

Mathematics Subject Classification

Keywords

Navigation