The geometry of rigid and non-rigid structures

Convex triangulated surfaces in three-space are rigid by Cauchy's Theorem. But what about non-convex surfaces? Some interesting recent examples of classes of non-convex surfaces have some convex-like properties, and yet are still rigid. On the other hand, in 1977, I constructed non-rigid (i.e. flexible) embedded surfaces. They have an unexpected property: while they flex, the volume bounded by these surfaces remains constant. In 1995, I. Sabitov proved this result, called the bellows property, which follows from the existence of a monic polynomial, with coefficients determined by the edge lengths, that is satisfied by the volume. But this may not be the end of the story; some conjectures may provide insight to the structure of flexible surfaces.