Are most jammed packings of dry, frictionless grains similar?

We study disordered jammed packings of frictionless disks interacting by either (one-sided) harmonic or Hertzian contact forces. We show that ensembles of such packings, in mechanical equilibrium, exhibit a very simple structure. In each case, we find that non-trivial fluctuations of the contact force network may be parameterised by a single dimensionless quantity for a large range of densities and disorder strengths. Surprisingly, a harmonic theory for displacement fluctuations in a crystal at thermal equilibrium is able to account for the statistics of these fluctuations.