Higher-Order Flexibility of Geometric Structures

A framework consists of knots, which are connected by bars according to a given combinatorial structure. By defining the lengths of the bars, the inner metric of the planar/spatial framework is fixed, but in general this assignment does not uniquely determine its embedding into the Euclidean plane/space; i.e. usually there exist several realizations. They result from solving a system of quadratic equations, which arise from the squared distances of related knots. A realization is referred to as rigid if it corresponds to an isolated solution of this system of equations. However, one calls a realization flexible (or shaky) of order n-1 if it relates to an n-fold solution of the system of equations; i.e. n>1 realizations coincide. But the determination of the flexes associated with higher-order flexible realizations is an open problem, which is the main focus of the research project. By considering two different realizations of a framework, which have an arbitrary relative pose to each other, the midpoints of corresponding knots can be used to generate a further framework with the same combinatorial structure (but with a different inner metric). It is known, that this averaging framework has a flexibility, which is in general of first order. A further aim of the research project is the generalization of this averaging technique towards the construction of higher-order flexible frameworks. Moreover, geometric structures should be systematically designed and analyzed having special properties regarding higher-order flexibility; for example mechanisms with standstill positions of arbitrary high order, which can be seen as generalizations of the well-known double-Watt mechanism of Connelly and Servatius. For reasons of demonstration we also plan to build corresponding models by means of 3D- printers. The research project will be conducted within the research group Differential Geometry and Geometric Structures of the Institute of Discrete Mathematics and Geometry at TU Wien, under the guidance of Georg Nawratil, who conceived and formulated all parts of the project proposal.