  We construct smooth BPS three-charge geometries that resolve the zero-entropy
singularity of the U(1) x U(1) invariant black ring. This singularity is
resolved by a geometric transition that results in geometries without any
branes sources or singularities but with non-trivial topology. These geometries
are both ground states of the black ring, and non-trivial microstates of the
D1-D5-P system. We also find the form of the geometries that result from the
geometric transition of N zero-entropy black rings, and argue that, in general,
such geometries give a very large number of smooth bound-state three-charge
solutions, parameterized by 6N functions. The generic microstate solution is
specified by a four-dimensional hyper-Kahler geometry of a certain signature,
and contains a ``foam'' of non-trivial two-spheres. We conjecture that these
geometries will account for a significant part of the entropy of the D1-D5-P
black hole, and that Mathur's conjecture might reduce to counting certain
hyper-Kahler manifolds.
