Speaker
Ben O'Connor
Abstract
Cauchy’s rigidity theorem states that if P and P′ are combinatorially equivalent convex polyhedra such that the corresponding facets of P and P′ are congruent, then P and P′ must be congruent polyhedra. For many years it was unknown whether the same theorem was true in general for non-convex polyhedron.
In 1977, more than 160 years after the work of Cauchy, Robert Connelly discovered a polyhedron P (without self-intersections) that allowed for a continuous deformation keeping the facets of P flat and congruent. It was soon noticed that the volume of Connelly’s polyhedron remained constant under the flexing motion, and the same fact was found to be true of all later-discovered flexible polyhedra. It was conjectured that this fact would hold in general, and it came to be known as the bellows conjecture.
In the period from 1995-1997, Idjad Sabitov proved the bellows conjecture by showing that for any (oriented) polyhedron P, the volume of P is a root of a polynomial depending only on the combinatorial structure and edge lengths of P. Moreover, the coefficients of the polynomial are themselves polynomials of the squares of the lengths of the edges of P with rational coefficients, with the coefficients depending only on the combinatorial structure of P. Hence, the volume of a polyhedron P with combinatorial structure K is a finitely valued function of the edge lengths of P, and in this way the theorem may be viewed as a generalization of Heron’s formula for the area of a triangle.
For more information, see the What If... ? seminar webpage.
