  There are deep, but hidden, geometric structures within jammed systems,
associated with hidden symmetries. These can be revealed by repeated
transformations under which these structures lead to fixed points. These
geometric structures can be found in the Voronoi tesselation of space defined
by the packing. In this paper we examine two iterative processes: maximum
inscribed sphere (MIS) inversion and a real-space coarsening scheme. Under
repeated iterations of the MIS inversion process we find invariant systems in
which every particle is equal to the maximum inscribed sphere within its
Voronoi cell. Using a real-space coarsening scheme we reveal behavior in
geometric order parameters which is length-scale invariant.
