Consider a closed finite triangulated oriented polyhedral surface S in
three-space. Regard the edges of S as rigid inextendible incompressible
bars at ideal universal joints, the vertices of S. There are several
examples when the bar constraints on the edges allow the shape of S to
change. In other words S has a non-trivial flex. We show that the volume
bounded by S during such a flex is constant. This can be though of as
saying that there is no exact mathematical "bellows" that can change its
enclosed volume.
This result (as well the notion of bounding volume) can be generalized
considerably to consider the case when S is any (possibly) singular
oriented simplicial cycle of dimension n-1 in any Euclidean
space of dimension n, where n is three or greater.
This is joint work with I. Sabitov and A. Walz.