Algebra versus analysis in the theory of flexible polyhedra

Details Algebra versus analysis in the theory of flexible polyhedra Algebra versus analysis in the theory of flexible polyhedra

2009-02-02

arxiv.org/abs/0902.0186v2

Aequationes Mathematicae, 79, no. 3 (2010), 229-235 and Erratum: Aequationes Mathematicae, 81, no. 1 (2011), 199

Mathematics

Metric Geometry

5 pages, 5 figures; condition (iii) in Theorem 5 is corrected

Scientific paper

10.1007/s00010-010-0024-3

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.

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