Algebra versus analysis in the theory of flexible polyhedra

作者：Alexandrov Victor^*

来源：Aequationes Mathematicae, 2010, 79(3): 229-235.

DOI：10.1007/s00010-010-0024-3

• Flexible polyhedron
• volume
• infinitesimal bending
• total mean curvature
• algebraic function

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. ex, 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.

• 出版日期2010-6