Algebra versus Analysis in the Theory of Flexible Polyhedra

Two basic theorems of the theory of flexible polyhedra were proven by completely different methods: R. Alexander used analysis, namely, the Stokes theorem, to prove that the total mean curvature remains constant during the flex, while I.Kh. 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 the 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. A polyhedron (more precisely, a polyhedral surface) is said to be flexible if its spatial shape can be changed continuously due to changes of its dihedral angles only, i. e., if every face remains congruent to itself during the flex. In other words, a polyhedron P0 is flexible if it is included in a continuous family {Pt}, 0 6 t 6 1, of polyhedra Pt such that, for every t, the corresponding faces of P0 and Pt are congruent while the polyhedra P0 and Pt are not congruent. The family {Pt}, 0 6 t 6 1, is called the flex of P0. Self-intersections are possible both for P0 and Pt provided the converse is not formulated explicitly. Without loss of generality we assume that the faces of the polyhedra are triangular. Flexible self-intersection free sphere-homeomorphic polyhedra in Euclidean 3space were constructed by R. Connelly thirty years ago [4], [6]. Since that time, various non-trivial properties of flexible polyhedra were discovered in the Euclidean 3-space [10] and 4-space [12] (for results in the hyperbolic 3-space, see [13]). Let us formulate two of them in a form suitable for our purposes. Let P be a closed oriented polyhedron in R, let E be the set of its edges, let |l| be the length of the edge l, and let α(l) be the dihedral angle of P at the edge l measured from inside of P . The sum