A Fractal Theory of Urban Growth

This paper presents an analytical framework for the physical environment of cities using fractal theory. The strength of the approach lies in its simplicity and precision. The equations presented in this article comprise: the number of occupied sites in an area; the population and the length of roads of a city; its fractal dimension; its number of average and maximum levels (floors per building); the average density of population and roads; what are the limits to growth as well as an analysis on some of the city's scaling laws. These equations describe to a high level of precision the real values measured in the system of the United Kingdom, for every city above 5,000 people, which amounts to a sample size of 1,031 cities. This work will allow further research into the nature of cities, since it enables the creation of synthetic cities, and further analytical derivations that can arise from these building blocks. The paper shows as well how the same set of equations can be used to characterise the internal distribution of cities from the perspective of its growth as a possible example of an application of the framework.