Path selection and growth in a Poisson field

The Poisson equation appears in many physical processes including electrostatics, fluid dynamics and fracture mechanics. Yet, pattern formation processes involving Poisson dynamics are not well-understood. Here we present a criterion for path selection for growing channels. We show that the growth of a channel in a Poisson field follows local symmetry in order to maximize the flux in its vicinity. We then use this criterion to reconstruct the history of a real network and to find the growth law associated with it. We also identify a cause for instability that results in a ramified structure in which the golden ratio prevails.