Isotropic growth on a grid

Isotropic growth from a single point on a two-dimensional square grid should generate an increasing sequence of discretized discs. We present a simple probabilistic model for growth on a grid, and discuss a class of parameterizations of the model (called kernels) which was conjectured (1) to produce isotropic growth. We disprove this conjecture, but we claim that these kernels produce growth that can be decomposed into isotropic and nonisotropic components. We also leave open the question of whether an isotropic parameterization exists. We review other probabilistic growth processes on grids, and describe qualitative and quantitative properties of the models. We also consider a deterministic growth model based on the diiusion equation, and show empirically that discretization of this model leads to a steady state connguration that appears to be polygonal. patterns The support of the Air Force OOce of Scientiic Research under Grant F49620-93-1-0039 is gratefully acknowledged. The authors are indebted to Dr. Murray Eden for his useful comments and suggestions.