  A disordered solid, such as an athermal jammed packing of soft spheres,
exists in a rugged potential-energy landscape in which there are a myriad of
stable configurations that defy easy enumeration and characterization.
Nevertheless, in three-dimensional monodisperse particle packings, we
demonstrate an astonishing regularity in the distribution of basin volumes. The
probability of landing randomly in a basin is proportional to its volume.
Ordering the basins according to their probability, $P(n)$, from the largest at
$n=1$ to smaller at larger $n$, we find approximately: $P(n) \propto n^{-1}$.
This order, persisting up to the largest systems for which we can collect
sufficient data, has implications for the dynamics of a system as it evolves
under perturbations. In monodisperse packings there is ``permutation symmetry''
since identical particles can always be interchanged without affecting the
system or its properties. Introducing any distribution of radii breaks this
symmetry and leads to a proliferation of distinct configurations. We present an
algorithm that partially restores permutation symmetry to such polydisperse
packings.
