Department of Mathematics, White Hall, Cornell University, Ithaca, New York, 14853-7901, USA, connelly@math.cornell.eduFaculty of Mechanics and Mathematics, Moscow State University Moscow, 119899, Russia, matan@top.rector.msu.su
Department of Mathematics,White Hall, Cornell University, Ithaca, New York, 14853-7901, USA, anke@math.cornell.edu
Abstract: Consider a closed finite triangulated oriented polyhedral surface $S$ in three-space. Regard the edges of $S$ as rigid inextendible incompressible bars attached 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.
The idea of the proof is to show that the volume satisfies a polynomial equation whose coefficients are themselves polynomials in the squares of the edge lengths of the polyhedron. Since the coefficients of this polynomial are a function only of the edge lengths, they remain same for all possible realizations of $S$. So there can only be a discrete finite number of values that are possible as a volume for those given edge lengths. Thus during a flex the volume bounded by $S$ remains constant.
Full text of the article: