  In computational models of particle packings with periodic boundary
conditions, it is assumed that the packing is attached to exact copies of
itself in all possible directions. The periodicity of the boundary then
requires that all of the particles' images move together. An infinitely
repeated structure, on the other hand, does not necessarily have this
constraint. As a consequence, a jammed packing (or a rigid elastic network)
under periodic boundary conditions may have a corresponding infinitely repeated
lattice representation that is not rigid or indeed may not even be at a local
energy minimum. In this manuscript, we prove this claim and discuss ways in
which periodic boundary conditions succeed to capture the physics of repeated
structures and where they fall short.
