Understanding the Fractal Dimensions of Urban Forms through Spatial Entropy

The spatial patterns and processes of cities can be described with various entropy functions. However, spatial entropy always depends on the scale of measurement, and it is difficult to find a characteristic value for it. In contrast, fractal parameters can be employed to characterize scale-free phenomena and reflect the local features of random multi-scaling structure. This paper is devoted to exploring the similarities and differences between spatial entropy and fractal dimension in urban description. Drawing an analogy between cities and growing fractals, we illustrate the definitions of fractal dimension based on different entropy concepts. Three representative fractal dimensions in the multifractal dimension set, capacity dimension, information dimension, and correlation dimension, are utilized to make empirical analyses of the urban form of two Chinese cities, Beijing and Hangzhou. The results show that the entropy values vary with the measurement scale, but the fractal dimension value is stable is method and study area are fixed; if the linear size of boxes is small enough (e.g., <1/25), the linear correlation between entropy and fractal dimension is significant (based on the confidence level of 99%). Further empirical analysis indicates that fractal dimension is close to the characteristic values of spatial entropy. This suggests that the physical meaning of fractal dimension can be interpreted by the ideas from entropy and scaling and the conclusion is revealing for future spatial analysis of cities.