Hellmuth Stachel - Between rigidity and flexibility

This talk was part of the Workshop “Rigidity and Flexibility of Geometric Structures” held September 24 -28, 2018 at the ESI.

The rigidity of structures is of importance for various areas of science such as chemistry, physics, biology, engineering, and material science. In this lecture the emphasis is on basic science rather than on applications. We focus on the geometry of various flexible or rigid structures such as polyhedra or frameworks, and we do not care about mechanics of materials or clearances of joints. We recall that a polyhedron is said to be flexible if its spatial form can be changed analytically with respect to a parameter, due to changes of its dihedral angles while every face remains congruent to itself during the flex. Conversely, it is said to be rigid if it admits only trivial deformations like translations and rotations. Besides, we can define an infinitesimal flexibility of structures, which lies somehow between flexibility and rigidity. Physical models of infinitesimally flexible structures show a slight mobility. We present several examples of structures where flexibility or infinitesimal flexibility is caused by particular dimensions, while in the generic case a structure of this type is rigid. Sometimes, a global or local symmetry causes the flexibility, but there are also overconstrained frameworks where the underlying algebraic conditions for flexibility have no relation to symmetry at all. In a few cases physical models will be provided to illustrate the presented results. These models are taken from the collection of kinematic models at the Institute of Discrete Mathematics and Geometry, TU Vienna.


 1. Lead 00:00:00
2. Four-bar linkages 00:00:22
3. Bipartite frameworks 00:23:05
4. Flexible polyhedra 00:38:53
5. Questions from the audience 00:52:32