Aequationes mathematicae

, Volume 79, Issue 3, pp 229–235 | Cite as

Algebra versus analysis in the theory of flexible polyhedra

  • Victor Alexandrov


Two basic theorems of the theory of flexible polyhedra were proven by completely different methods: Alexander used analysis, namely, the Stokes theorem, to prove that the total mean curvature remains constant during the flex, while Sabitov used algebra, namely, the theory of resultants, to prove that the oriented volume remains constant during the flex. We show that none of these methods can be used to prove both theorems. As a by-product, we prove that the total mean curvature of any polyhedron in the Euclidean 3-space is not an algebraic function of its edge lengths.

Mathematics Subject Classification (2000)

Primary 52C25 Secondary 51M20 


Flexible polyhedron volume infinitesimal bending total mean curvature algebraic function 


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© Springer Basel AG 2010

Authors and Affiliations

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

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