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Algebra versus analysis in the theory of flexible polyhedra. (English) Zbl 1220.52005
Summary: 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.
Erratum: Condition (iii) in Theorem 5 is formulated incorrectly. It is vital to the understanding that it should read as follows:
(iii) \(P\) contains no vertex \(V\) such that some three edges of \(P\) incident to \(V\) lie in a plane.

MSC:
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
51M20 Polyhedra and polytopes; regular figures, division of spaces
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