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].
Similar content being viewed by others
References
Trushkina V. I., “A coloring theorem and the rigidity of a convex polyhedron,” Ukrain. Geom. Sb., No. 24, 116–122 (1981).
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.
Trushkina V. I., “Method of 3-coloring of graphs,” Siberian Math. J., 38, No. 2, 331–342 (1987).
Connelly R., “Rigidity,” in: Handbook of Convex Geometry. Vol. A, North-Holland, Amsterdam, 1993, pp. 223–271.
Alexandrov A. D., Convex Polyhedra, Springer, Berlin (2005).
Berger M., Géométrie. Vol. 1–5, Nathan, Cedic, Paris (1977).
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.
Whiteley W., “Rigidity and scene analysis,” in: Handbook of Discrete and Computational Geometry, CRC Press, Boca Raton, FL, 1997, pp. 893–916.
Alexandrov V., “Inverse function theorems and their applications to the theory of polyhedra, ” Rev. Math. Math. Phys. (to appear).
Pak I., Lectures on Combinatorial Geometry and Convex Polytopes (a monograph in preparation).
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.
Gluck H., “Almost all simply connected closed surfaces are rigid,” in: Lecture Notes in Math., Springer, Berlin, 438, 1975, pp. 225–239.
Roth B., “Rigid and flexible frameworks,” Amer. Math. Monthly, 88, No. 1, 6–21 (1981).
Whiteley W., “Infinitesimally rigid polyhedra. I. Statics of frameworks,” Trans. Amer. Math. Soc., 285, No. 2, 431–465 (1984).
Author information
Authors and Affiliations
Additional information
Original Russian Text Copyright © 2006 Pak Igor
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 4, pp. 859–864, July–August, 2006.
Rights and permissions
About this article
Cite this article
Pak, I. A short proof of rigidity of convex polytopes. Sib Math J 47, 710–713 (2006). https://doi.org/10.1007/s11202-006-0081-y
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11202-006-0081-y