On the flexibility of tessellations

A bar-joint framework, which is a (possibly countably infinite) graph together with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the distances between adjacent vertices, otherwise it is called rigid. In general, deciding rigidity/flexibility of a given finite framework is NP-hard, but there are more tractable subclasses. For instance Bolker and Crapo showed that rigidity of a braced finite grid of squares is determined by connectedness of an auxiliary bipartite graph. We focus on finite and infinite frameworks that "consist of triangles and parallelograms", for which a certain equivalence relation defined on the set of edges can be used to decide flexibility. We illustrate the result on the 1-skeleta of plane tessellations. We shall also discuss the rotationally symmetric case.