Advertisement

Siberian Mathematical Journal

, Volume 47, Issue 4, pp 710–713 | Cite as

A short proof of rigidity of convex polytopes

  • Igor Pak
Article
  • 61 Downloads

Abstract

We present a much simplified proof of Dehn’s theorem on the infinitesimal rigidity of convex polytopes. Our approach is based on the ideas of Trushkina [1] and Schramm [2].

Keywords

convex polytope infinitesimal rigidity Dehn theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Trushkina V. I., “A coloring theorem and the rigidity of a convex polyhedron,” Ukrain. Geom. Sb., No. 24, 116–122 (1981).Google Scholar
  2. 2.
    Schramm O., “How to cage an egg,” Invent. Math., 107, No. 3, 543–560 (1992). Available at http://dz-srv1.sub.unigoettingen.de/cache/toc/D183703.html.zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Trushkina V. I., “Method of 3-coloring of graphs,” Siberian Math. J., 38, No. 2, 331–342 (1987).CrossRefGoogle Scholar
  4. 4.
    Connelly R., “Rigidity,” in: Handbook of Convex Geometry. Vol. A, North-Holland, Amsterdam, 1993, pp. 223–271.Google Scholar
  5. 5.
    Alexandrov A. D., Convex Polyhedra, Springer, Berlin (2005).Google Scholar
  6. 6.
    Berger M., Géométrie. Vol. 1–5, Nathan, Cedic, Paris (1977).Google Scholar
  7. 7.
    Dehn M., “Über die Starreit konvexer Polyeder,” Math. Ann., Bd 77, 466–473 (1916). Available at http://dz-srv1.sub.unigoettingen.de/cache/toc/D37460.html.zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Whiteley W., “Rigidity and scene analysis,” in: Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, FL, 1997, pp. 893–916.Google Scholar
  9. 9.
    Alexandrov V., “Inverse function theorems and their applications to the theory of polyhedra, ” Rev. Math. Math. Phys. (to appear).Google Scholar
  10. 10.
    Pak I., Lectures on Combinatorial Geometry and Convex Polytopes (a monograph in preparation).Google Scholar
  11. 11.
    Fedorchuk M. and Pak I., “Rigidity and polynomial invariants of convex polytopes,” Duke Math. J., 129, 371–404 (2005). Available at http://www-math.mit.edu/∼pak.zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gluck H., “Almost all simply connected closed surfaces are rigid,” in: Lecture Notes in Math., Springer, Berlin, 438, 1975, pp. 225–239.Google Scholar
  13. 13.
    Roth B., “Rigid and flexible frameworks,” Amer. Math. Monthly, 88, No. 1, 6–21 (1981).zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Whiteley W., “Infinitesimally rigid polyhedra. I. Statics of frameworks,” Trans. Amer. Math. Soc., 285, No. 2, 431–465 (1984).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Igor Pak
    • 1
  1. 1.Massachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations