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Journal of Geometry

, 110:38 | Cite as

A sufficient condition for a polyhedron to be rigid

  • Victor AlexandrovEmail author
Article

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.

Keywords

Flexible polyhedron Dihedral angle Dehn invariant Hamel basis Bricard octahedron 

Mathematics Subject Classification

Primary 52C25 Secondary 52B70 51M20 

Notes

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.

Compliance with ethical standards

Conflict of interest

The author states that he does not have any conflicts of interest to declare.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Department of PhysicsNovosibirsk State UniversityNovosibirskRussia

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