%0 Journal Article
%T Characterizing Growth and Form of Fractal Cities with Allometric Scaling Exponents
%A Yanguang Chen
%J Discrete Dynamics in Nature and Society
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/194715
%X Fractal growth is a kind of allometric growth, and the allometric scaling exponents can be employed to describe growing fractal phenomena such as cities. The spatial features of the regular fractals can be characterized by fractal dimension. However, for the real systems with statistical fractality, it is incomplete to measure the structure of scaling invariance only by fractal dimension. Sometimes, we need to know the ratio of different dimensions rather than the fractal dimensions themselves. A fractal-dimension ratio can make an allometric scaling exponent (ASE). As compared with fractal dimension, ASEs have three advantages. First, the values of ASEs are easy to be estimated in practice; second, ASEs can reflect the dynamical characters of system's evolution; third, the analysis of ASEs can be made through prefractal structure with limited scale. Therefore, the ASEs based on fractal dimensions are more functional than fractal dimensions for real fractal systems. In this paper, the definition and calculation method of ASEs are illustrated by starting from mathematical fractals, and, then, China's cities are taken as examples to show how to apply ASEs to depiction of growth and form of fractal cities. 1. Introduction Dimension is a measurement of space, and measurement is the basic link between mathematical models and empirical research. So, dimension is a necessary measurement for spatial analysis. Studying geographical spatial phenomena of scaling invariance such as cities and systems of cities has highlighted the value of fractal dimension [1¨C6]. However, there are two problems in practical work. On the one hand, sometimes it is difficult for us to determine the numerical value of fractal dimension for some realistic systems, but it is fairly easy to calculate the ratio of different fractal parameters. On the other hand, in many cases, it is enough to reveal the systemĄ¯s information by the fractal-dimension ratios and it is unnecessary to compute fractal dimension further [7]. The ratio of different dimensions of a fractal can constitute an allometric coefficient under certain conditions. As a parameter of scale-free systems, the allometric coefficient is in fact a scaling exponent, and the fractal-dimension ratio can be called allometric scaling exponent (ASE). The use of ASEs for the simple regular fractals in the mathematical world is not very noticeable. But for the quasifractals or random fractals in the real world, the function of ASEs should be viewed with special respect. A city can be regarded as an evolutive fractal. The land-use
%U http://www.hindawi.com/journals/ddns/2010/194715/