Algebra versus analysis in the theory of flexible polyhedra

TL;DR

This paper compares algebraic and analytical methods in the theory of flexible polyhedra, showing their limitations and proving that certain geometric quantities are not algebraic functions of edge lengths.

Contribution

It demonstrates that algebraic and analytical approaches cannot prove both key theorems simultaneously and establishes that total mean curvature is not an algebraic function of edge lengths.

Findings

Analytical methods prove total mean curvature is constant during flex.

Algebraic methods prove volume remains constant during flex.

Total mean curvature is not an algebraic function of edge lengths.

Abstract

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.