Chapter 9: Scaling, fractals and the spatial complexity of cities
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Geographical phenomena of cities fall into two categories: one is simple systems which characteristic scales, and the other is complex systems without characteristic scale. The key of conventional mathematical modeling and quantitative analysis is to find characteristic length. The scale-free phenomena of cities cannot be effectively characterized by traditional mathematical methods. Thus the characteristic scale-based spatial analysis should be replace by scaling-based spatial analysis. Based on mathematical reasoning and empirical studies, this work is devoted to discussing scaling, fractals, spatial complexity of cities and the inherent relationships between these theories. The main points are as follows. First, scaling is one of basic properties of cities, and the essence of scaling in cities is invariance of contraction or dilation transform of urban models. Second, fractal pattern is the spatial order emerging from urban self-organized evolution. The essence of city fractals is scaling symmetry. Third, spatial complexity of cities is related to scaling and fractals. The ideas from scaling and fractals make new ways for understanding complex spatial systems. In conclusions, fractal geometry provides a powerful tool for scaling analysis, and can be used to explore spatial complexity and singularity of cities. Scaling and fractals can be integrated into a new theoretical framework of spatial complexity of cities.

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