Louis Theran, Aalto University

Abstract:

Bar-joint frameworks are mechanical structures made of fixed-length bars, connected by universal joints with full rotational freedom. The allowed motions preserve the lengths and connectivity of the bars. The rigidity question for bar-joint frameworks asks if all the allowed motions are rigid body motions. For frameworks with generic geometry in dimension 2, a theorem of Laman classifies the minimally rigid combinatorial types (i.e., the graph that has as its edges the bars). Laman’s result is proved by making rigorous a degree of freedom heuristic that goes back to Maxwell. I’ll discuss similar types of results for symmetric frameworks (which are far from generic), with a focus on the appropriate analogue of Maxwell counting.