In diffusion-limited aggregation (DLA), particles undergo random walks due to Brownian motion. They cluster together to form aggregates.
DLA can be used to model systems such as lichen growth, the generation of polymers out of solutions, carbon deposits on the walls of a cylinder of a Diesel engine, path of electric discharge, and urban settlement.
In this simulation, the initial aggregate is set to be the bottom row of an NxN lattice. Particles are launched from a random cell in the top row. A particle's random walk is set to have the following probabilities: up: 0.15, down: 0.35, and left or right: 0.25. The particle continues until it sticks to a neighbouring cell or leaves the lattice.
DLAs appear to be related to the Fibonacci sequence in terms of the branching sequence of the aggregate (see the references for more information about this, I haven't looked into it in much detail). Additionally, small clusters evolve toward a five-branch symmetry. Note that the ratio of successive Fibonacci numbers approaches the golden ratio, which appears in the geometry of the pentagon.