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Polyhedra with bendings and Riemann surfaces

The history of the conjecture that closed polyhedra do not admit bendings goes back many years. It is now known that almost no polyhedra admit bendings; however, counterexamples have been constructed, as well. In 1897 Bricard [1] listed all octahedra (six vertices) with bendings, but none of them turned out to be immersed. Only in 1975 did Connelly [2] succeed in constructing an immersed polyhedron with bendings, having 12 vertices, and in 1977 an embedded one (with 18 vertices). Finally, in 1978 Steffen constructed an embedded polyhedron with bendings, having 9 vertices (see [3], [4]). However, the problem of the minimal number of vertices of such polyhedra has remained open. We prove that embedded polyhedra with fewer vertices do not admit bendings. The proof of this fact can be reduced to the study of bendable polyhedra of a specific combinatorial type on the basis of the study of some Riemann surfaces appearing in the analytic continuation of the equations of bending to a complex domain. A similar approach was used by Connelly [5] in proving that pendants (bipyramids) do not admit bendings. However, in our case the corresponding Riemann surfaces are needed as domains of existence of some other functions. The method offered below enables us to produce a whole new class of polyhedra without bendings. Let ft" be a two-dimensional closed simplicial complex, homeomorphic to the sphere, and let E be the three-dimensional Euclidean space. Definition 1. A polyhedron is a globally continuous map Ρ : Κ —+ Ε that is linear on each simplex.