## The geometry of rigid and non-rigid structures

- Date: 10/10/2007

Robert Connelly (Cornell University)

University of Calgary

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.

PIMS Distinguished Lecture 2007