  Plant differently colored points in the plane, then let random points
("Poisson rain") fall, and give each new point the color of the nearest
existing point. Previous investigation and simulations strongly suggest that
the colored regions converge (in some sense) to a random partition of the
plane. We prove a weak version of this, showing that normalized empirical
measures converge to Lebesgue measures on a random partition into measurable
sets. Topological properties remain an open problem. In the course of the
proof, which heavily exploits time-reversals, we encounter a novel self-similar
process of coalescing planar partitions. In this process, sets $A(z)$ in the
partition are associated with Poisson random points $z$, and the dynamics are
as follows. Points are deleted randomly at rate $1$, when $z$ is deleted, its
set $A(z)$ is adjoined to the set $A(z^\prime)$ of the nearest other point
$z^\prime$.
