Paradoxical flexibility: frameworks and polyhedra

Abstract

One of the main questions of rigidity theory is whether a bar-joint framework, which is a graph with a realization of its vertices in the d-dimensional space, allows a continuous deformation preserving the distances between adjacent vertices. If yes, the framework is called flexible, otherwise rigid. For a fixed graph, either all generic frameworks are rigid, or all generic ones are flexible. However, non-generic realizations might behave differently yielding for instance paradoxical motions. A few years ago, we have characterized the existence of a (non-generic) flexible realization in the plane for a given graph in terms of special edge colorings, called NAC-colorings. In this talk, this surprising interplay between combinatorics and geometry and its various extensions shall be presented. We focus also on polyhedra with triangular faces which can be considered as bar-joint frameworks in the 3-space. In particular, a new result on the smallest flexible polyhedron without self-intersections shall be given.