ABSTRACT
It is important to study the impact of symmetry on the rigidity and flexibility properties of geometric constraint systems. This chapter presents some fundamental methods and results for the detection of symmetry-induced infinitesimal flexes and self-stresses in frameworks which count to be isostatic without symmetry. One may also seek combinatorial characterizations of infinitesimally rigid periodic frameworks on a more basic level. The chapter provides some fundamental results concerning the rigidity of infinite periodic frameworks, both with a fixed and a flexible lattice representation. Combinatorial characterizations of infinitesimally rigid crystallographic frameworks in the plane with a fully flexible lattice representation are given in for the case where the group is generated by translations and rotations. There exist several initial results regarding the rigidity of periodic frameworks with additional symmetry which are forced to maintain the full crystallographic group of the framework throughout any motion.
