Counting of freedom in symmetric frameworks
Louis Theran, Aalto University
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.