Paradoxical flexibility of frameworks

Paradoxical motions occur when some structure that is expected to be rigid turns out to be flexible for some very particular cases. As a basic real world structure we may think of truss constructions like bridges, towers or scaffolds. Of course we do want them to be stable. These constructions have a relatively simple structure, so we know how to build them in a rigid way. Let us consider more general but also more abstract constructions. We take a set of rotational joints that are connected by some bars with fixed lengths. When we build a triangle out of them and place it in the plane, then the only way to move it is to rotate or translate the whole triangle, or to do a combination of those. Such an object we call rigid. When we take four joints with four bars connected in a cycle, then we get something flexible, because the object allows a continuous deformation preserving the length of the bars. If we forget about the lengths of the bars we get a graph. This is a set of vertices (representing the joints) and edges (representing the bars). It turns out that in general the structure of a graph already tells a lot about its rigidity properties. But many graphs yield a rigid object for almost all choices of edge lengths but just for some very particular ones they turn out to result in a flexible one. In this project we investigate these special instances. It is known that flexible situations can be related to special colorings of the edges. However, it is complicated to find such colorings. Part of the project therefore deals with speeding up this computation. Furthermore we investigate symmetric graphs and paradoxical motions preserving symmetry. While there is a connection between colorings and rotationally symmetric motions, no such result is yet known for other symmetries. Also in other applications rigidity plays a role. For instance in sensor networks the measurements of distances can be used for determining positions of sensors. But this can only be done if for the underlying graph there is only one placement up to rotations and translations that gives the measured lengths. But also such graphs can have paradoxical flexibility, which would make it impossible to detect the positions. Finally, there are graphs that are generically flexible but certain placements allow additional mobility. We can also add different constraints on the graph or the motions. In all such cases paradoxical flexibility is interesting and is investigated in this project.