  Isostatic frames are mechanical networks that are simultaneously rigid and
free of self-stress states, and is a powerful concept in understanding phase
transitions in soft matter and designing of mechanical metamaterials. Here we
analyze substructures of isostatic frames, by generalizing ``Assur graphs'' to
the torus and examine them in physical systems. We show that the contact
network of marginally jammed packings approach torus Assur graphs in the
thermodynamic limit, and demostrate how Assur graphs offer a new design
principle for mechanical metamaterials in which motion and stress can propagate
in reconfigurable pathways, while rigidity of the entire structure is
maintained.
