Linear bendings of star-like bipyramids

This paper is a revised and an extended version of the article [A.D. Milka, Linear bending of star-like pyramids, Comptes Rendus Mecanique 331 (12) (2003) 805–810]. A family of polyhedra possessing unusual deformation properties is found. On one hand, models of these polyhedra admit free continuous large reversible bendings without visible distortions of the material. On the other hand, the polyhedra themselves are mathematically rigid and do not admit continuous bendings in the sense of O. Cauchy. The found polyhedra are called model flexors in order to distinguish them from theoretical flexors of R. Connelly. Bendings of the models are asymptotically exactly approximated by linear bendings of polyhedra. They represent a non-rigid, soft or retarded, loss of stability which corresponds to the loss of stability “in the small” in the sense of L. Euler. This new phenomenon in mechanics of deformable solid bodies may be considered as an original geometric catastrophe machine which supplements known physical models by E.C. Ziman and T. Poston.