  Using a previously developed experimental method to reduce friction in
mechanically stable packings of disks, we find that frictional packings form
tree-like structures of geometrical families that lie on reduced dimensional
manifolds in configuration space. Each branch of the tree begins at a point in
configuration space with an isostatic number of contacts and spreads out to
sequentially higher dimensional manifolds as the number of contacts are
reduced. We find that gravitational deposition of disks produces an initially
under-coordinated packing stabilized by friction on a high-dimensional
manifold. Using short vibration bursts to reduce friction, we compact the
system through many stable configurations with increasing contact number and
decreasing dimensionality until the system reaches an isostatic frictionless
state. We find that this progression can be understood as the system moving
through the null-space of the rigidity matrix defined by the interparticle
contact network in the direction of the gravitational force. We suggest that
this formalism can also be used to explain the evolution of frictional packings
under other forcing conditions.
