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A sufficient condition for a polyhedron to be rigid

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Abstract

We study oriented connected closed polyhedral surfaces with non-degenerate triangular faces in three-dimensional Euclidean space, calling them polyhedra for short. A polyhedron is called flexible if its spatial shape can be changed continuously by changing its dihedral angles only. We prove that for every flexible polyhedron some integer combination of its dihedral angles remains constant during the flex. The proof is based on a recent result of A. A. Gaifullin and L. S. Ignashchenko.

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Acknowledgements

The author is grateful to Alexander A. Gaifullin and Idzhad Kh. Sabitov for their comments on a preliminary version of this article and to an anonymous referee for several useful comments and constructive suggestions.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Koptyug ave., 4, Novosibirsk, Russia, 630090

    Victor Alexandrov

  2. Department of Physics, Novosibirsk State University, Pirogov str., 2, Novosibirsk, Russia, 630090

    Victor Alexandrov

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  1. Victor Alexandrov
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Correspondence to Victor Alexandrov.

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Dedicated to Academician Yuriĭ Grigor’evich Reshetnyak on the occasion of his 90th birthday.

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Alexandrov, V. A sufficient condition for a polyhedron to be rigid. J. Geom. 110, 38 (2019). https://doi.org/10.1007/s00022-019-0492-0

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  • DOI: https://doi.org/10.1007/s00022-019-0492-0

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