Infinitesimally rigid polyhedra. I. Statics of frameworks

Abstract:

From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation of the natural faces on these vertices, such polyhedra are infinitesimally rigid. In this paper the dual (and equivalent) concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex (and other) polyhedra. The static techniques introduced provide a new simplified proof of Alexandrov’s theorem, as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces, including frameworks with vertices interior to the faces. The static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron (nonconvex, toroidal, etc.). The techniques are also applied to derive the static rigidity of tensegrity frameworks (with cables and struts in place of bars), and the static rigidity of frameworks projectively equivalent to known polyhedral frameworks. Finally, as an exercise to give an additional perspective to the results in $3$-space, detailed analogues of Alexandrov’s theorem are presented for convex $4$-polytopes built as bar and joint frameworks in $4$-space.