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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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##### References:
 [1] Ahlfors L.: Complex Analysis, 3rd edn. McGraw-Hill, New York (1979) · Zbl 0395.30001 [2] Alexander R.: Lipschitzian mappings and total mean curvature of polyhedral surfaces, I. Trans. Am. Math. Soc. 288, 661–678 (1985) · Zbl 0563.52008 [3] Almgren, F., Rivin, I.: The mean curvature integral is invariant under bending. In: The Epstein Birthday Schrift, University of Warwick, pp. 1–21 (1998) · Zbl 0914.53007 [4] Connelly, R.: Conjectures and open questions in rigidity. In: Proc. Int. Congr. Math., Helsinki 1978, vol. 1, pp. 407–414 (1980) [5] Connelly R., Sabitov I., Walz A.: The bellows conjecture. Beitr. Algebra Geom. 38, 1–10 (1997) · Zbl 0939.52009 [6] Kuiper, N.H.: Sphères polyedriques flexibles dans E 3, d’après Robert Connelly. In: Seminaire Bourbaki, vol. 1977/78, Expose No. 514, Lect. Notes Math., vol. 710, pp. 147–168 (1979) · Zbl 0435.53043 [7] Sabitov, I.Kh.: Local theory on bendings of surfaces. In: Geometry III. Theory of Surfaces. Encycl. Math. Sci., vol. 48, pp. 179–250 (1992) [8] Sabitov I.Kh.: The volume of a polyhedron as a function of its metric (in Russian). Fundam. Prikl. Mat. 2, 1235–1246 (1996) · Zbl 0904.52002 [9] Sabitov I.Kh.: The volume as a metric invariant of polyhedra. Discrete Comput. Geom. 20, 405–425 (1998) · Zbl 0922.52006 [10] Schlenker, J.-M.: La conjecture des soufflets (d’après I. Sabitov). In: Seminaire Bourbaki, vol. 2002/03. Société Math. de France, Paris. Astérisque 294, 77–95, Exp. No. 912 (2004) [11] Souam R.: The Schläfli formula for polyhedra and piecewise smooth hypersurfaces. Differ. Geom. Appl. 20, 31–45 (2004) · Zbl 1065.52015 [12] Stachel H.: Flexible cross-polytopes in the Euclidean 4-space. J. Geom. Graph. 4, 159–167 (2000) · Zbl 0977.52028 [13] Stachel H. et al.: Flexible octahedra in the hyperbolic space. In: Prékopa, A. (eds) Non-Euclidean Geometries. János Bolyai memorial volume, pp. 209–225. Springer, New York (2006) · Zbl 1100.52005
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