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Using a previously developed experimental method to reduce friction inmechanically stable packings of disks, we find that frictional packings formtree-like structures of geometrical families that lie on reduced dimensionalmanifolds in configuration space. Each branch of the tree begins at a point inconfiguration space with an isostatic number of contacts and spreads out tosequentially higher dimensional manifolds as the number of contacts arereduced. We find that gravitational deposition of disks produces an initiallyunder-coordinated packing stabilized by friction on a high-dimensionalmanifold. Using short vibration bursts to reduce friction, we compact thesystem through many stable configurations with increasing contact number anddecreasing dimensionality until the system reaches an isostatic frictionlessstate. We find that this progression can be understood as the system movingthrough the null-space of the rigidity matrix defined by the interparticlecontact network in the direction of the gravitational force. We suggest thatthis formalism can also be used to explain the evolution of frictional packingsunder other forcing conditions.

Experiments demonstrate that the null space of the rigidity matrix determines grain motion during vibration-induced compaction