Generating Urban Forms from Diffusive Growth

In this paper, a formal model of diffusion-limited aggregation is presented which can be used to generate a continuum of urban forms from linear to concentric. A brief derivation of this continuum model is presented in terms of its relationship to potential theory, and a method for its solution is outlined. The forms produced by the model are subject to fractional power laws which relate the occupancy of sites and densities to distances. Fractal dimensions can be derived from these fractional powers, and conventional and fast methods of estimation are introduced. A large-scale simulation provides the baseline for comparison and its average fractal dimension is estimated as 1.701 ± 0.025 from thirty aggregates. The effect of constraining the physical limits of the lattice on which such aggregates are grown is explored and the model is then used to generate a continuum of forms determined by a control parameter which distorts the potential field of the model. A series of forms from the linear with fractal dimension close to 1 to the concentric with a fractal dimension near to 2, are grown. The model is then applied to urban development in the town of Cardiff and various simulations are attempted which point the way to further research.