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.

Other Information: 

PIMS Distinguished Lecture 2007