  This paper presents a range of results in partial differential equations
(PDEs) in which Voronoi patterns arise. We investigate the connection between
the solution to an elliptic equation and its probabilistic interpretation as a
stochastic colonization game. An agent-based model is designed and implemented
to generate the Voronoi cells simulating experimental results with bacteria. We
also consider the analytical solution to the problem, which enables us to
define what we call a harmonic Voronoi tessellation. We analyze parabolic
equations in Riemannian manifolds, which have important applications in
chemical reactions and diffusive fronts. By utilizing short-time heat kernel
estimates, we demonstrate that the interaction of $n$ point sources gives rise
to a Voronoi tessellation. We recall some well-known results of wavefronts
interactions from point light sources and the Huygens principle. We apply
results about the particular set of weak solutions to the eikonal equation to
characterize Voronoi patterns arising in this context as rectifiable sets.
Finally, we present an optimal transport problem and the corresponding
Monge-Amp\`ere equation, in which a uniform measure is transported to a sum of
$n$ Dirac masses with a cost given by the Euclidean distance. These problems
are naturally linked to power sets, a generalization of Voronoi tessellations.
