The Size, Shape and Dimension of Urban Settlements

In this paper, we propose a scale theory of urban form and growth which enables us to consistently explain and estimate relationships between urban population size, area, field and boundary length for a system of settlements. Our approach is based on a synthesis of allometry and fractal growth theory, and the associated relationships are uniquely specified by dimensional parameters whose values vary from 1 to 2, from the line to the plane. The theory assumes that the form of settlements is tentacular and that the population density of these forms is constant with respect to their size. After the theory has been presented, four relationships - two allometric, relating populations and boundaries (or envelopes) to urban areas, and two fractal, relating the same variables to the urban field size - are estimated for some 70 settlements which compose the urban system in the English County of Norfolk. The hypothesized values of the dimensions characterizing these four relationships are confirmed by regression estimates and these results are given further strength when the same relations are re-estimated for various subsets of settlements in the Norfolk urban system. We conclude that the geometric form of the settlements system is consistent with the model we have adopted, that population density is constant at all scales, and that urban boundaries have a degree of irregularity measured by a fractal dimension similar to that conventionally assumed for coastlines. Finally, we suggest directions for further research.