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Two basic theorems of the theory of flexible polyhedra were proven bycompletely different methods: R. Alexander used analysis, namely, the Stokestheorem, to prove that the total mean curvature remains constant during theflex, while I.Kh. Sabitov used algebra, namely, the theory of resultants, toprove that the oriented volume remains constant during the flex. We show thatnone of these methods can be used to prove the both theorems. As a by-product,we prove that the total mean curvature of any polyhedron in the Euclidean3-space is not an algebraic function of its edge lengths.

Algebra versus analysis in the theory of flexible polyhedra