Divine Proportions and the Fractal Dimension of DLCA Aggregates

Aggregation is a non-equilibrium process of fundamental importance for all dispersed particulate systems. Here I present a restricted hierarchical model of diffusion limited cluster-cluster aggregation (DLCA). The model yields an analytical calculation of the fractal dimensions and self-preserving cluster shapes in two and three spatial dimensions in excellent agreement with those found in nature and simulations. Remarkably, the shape is described by the Fibonacci series and the divine proportion in two dimensions and d-dimensional generalization of the Fibonacci series and the divine proportion in three and higher dimensions.