  We propose that every supersymmetric four dimensional black hole of finite
area can be split up into microstates made up of primitive half-BPS "atoms''.
The mutual non-locality of the charges of these "atoms'' binds the state
together. In support of this proposal, we display a class of smooth,
horizon-free, four dimensional supergravity solutions carrying the charges of
black holes, with multiple centers each carrying the charge of a half-BPS
state. At vanishing string coupling the solutions collapse to a bound system of
intersecting D-branes. At weak coupling the system expands into the non-compact
directions forming a topologically complex geometry. At strong coupling, a new
dimension opens up, and the solutions form a "foam'' of spheres threaded by
flux in M-theory. We propose that this transverse growth of the underlying
bound state of constitutent branes is responsible for the emergence of black
hole horizons for coarse-grained observables. As such, it suggests the link
between the D-brane and "spacetime foam'' approaches to black hole entropy.
