ABSTRACT
Since symmetry is ubiquitous in both man-made structures (e.g., buildings or mechanical linkages) and in structures found in nature (e.g., proteins or crystals), it is natural to consider the impact of symmetry on the rigidity and flexibility properties of frameworks. The special geometry induced by various symmetry groups often leads to added first-order (and sometimes even continuous) flexibility in the structure. These phenomena have been studied in the following two settings:
Forced Symmetry: The framework is symmetric (with respect to a certain group) and must maintain this symmetry throughout its motions.
Incidental Symmetry: The framework is symmetric (with respect to a certain group), but is allowed to move in unrestricted ways.
The key tool for analyzing the forced-symmetric rigidity properties of a symmetric framework is its corresponding group-labeled quotient graph (or “gain graph”). In particular, using very simple counts on the number of vertex and edge orbits under the group action (i.e., vertices and edges of the gain graph), we can often detect symmetry-preserving first-order flexibility in symmetric frameworks that are generically rigid without symmetry. For configurations which are regular modulo the given symmetry, these first-order flexes even extend to continuous flexes. By introducing (gain-)sparsity counts for all subgraphs of a gain graph, Laman-type combinatorial characterizations of all (symmetry-)regular forced-symmetric rigid frameworks have also been obtained for various symmetry groups. Moreover, these combinatorial results have been extended to “body-bar frameworks” with an arbitrary symmetry group in d-dimensional space.