Form Follows Function: Reformulating Urban Population Density Functions

In this paper, we argue that the most appropriate form for urban population density models is the inverse power function, contrary to conventional practice, which is largely based upon the negative exponential. We first show that the inverse power function has several theoretical properties which have hitherto gone unremarked in the literature. Our main argument, however, is based on the notion that a density function should describe the extent to which the space available for urban development is filled. To this end, we introduce ideas from urban allometry and fractal geometry to demonstrate that the inverse power model is the only function which embodies the fractal property of self-similarity which we consider to be a basic characteristic of urban form and density. In short, we show that the distance parameter a of the inverse power model is a measure of the extent to which space is filled, and that its value is determined by the basic relation D+α=2 where D is the fractal dimension of the city in question. We then test this model using four data sets which measure the density and morphology of the city of Seoul. Using a variety of estimation methods such as loglinear regression, dimensional approximation and entropy-maximising, we estimate dimension and density parameters for 136 variants of the function and its data sets. From these estimates, 125 are within the values hypothesised and this suggests fairly conclusively that the value of the density parameter a for the inverse power function should be within the range 0-1 and probably between 0.2 and 0.5. Many related questions are raised by this analysis which will form the subject of future research.