Bellows conjecture

Builds on Flexible polyhedron Polyhedron

Statement

Let $P_t$ , for $t \in [0, 1]$ , be a continuous flex of a closed orientable polyhedral surface in $\R^3$ . That is, each face of $P_t$ stays congruent to the same face of $P_0$ , and the combinatorics stay the same. Then the oriented volume enclosed by $P_t$ does not depend on $t$ .

Remarks

The statement was proved in 1997 by Connelly, Sabitov and Walz, so it is now a theorem. The proof rests on a result of Sabitov: the volume of a polyhedron is a root of a monic polynomial whose coefficients are polynomials in the squared edge lengths. Along a flex the edge lengths are fixed, so the volume takes only finitely many values. Since it varies continuously in $t$ , it is constant.

Example: every flexible polyhedron, such as Steffen's polyhedron, keeps its volume while it moves. So such a polyhedron cannot work as a bellows that pumps air by changing its volume.

Going deeper

For a student

The result complements Cauchy's rigidity theorem. Cauchy rules out any flex of a convex polyhedron, while the bellows theorem allows flexes of nonconvex surfaces but fixes their volume. Its algebraic proof treats the volume as an algebraic function of the edge lengths, a three-dimensional relative of Heron's formula.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Tag	`BG3XQF6V58`
URI	`m2://BG3XQF6V58`
Kind	statement, theorem
Writing	W0 stub
MSC	52C25, 51M25, 52B10
Status	This copy of the site has no build data. Run `m2 status BG3XQF6V58`.
About	Flexible polyhedron
Uses	Flexible polyhedron, Polyhedron