DLCA Aggregate Divine Proportion Shape, Fractal Dimension, and the Pascal Triangle.

Aggregation is a non-equilibrium process of fundamental importance for 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. It is shown that the shape is described by a generalization of the Fibonacci series to all spatial dimensions and found in systematic diagonals of the Pascal Triangle.