Studies On Spatial Complexity Of Fractal Urban Systems

The spatial complexity of urban systems and complication in process of urbanization are studied using the postmodern mathematics of fractal and chaos and some idea from complexity theory in order to bring to light the complex spatial dynamics of self-organizing networks of cities. All the contents are developed around a set of scaling laws taking the form of exponentials which relate to almost all the issues of complexity including fractals, chaos, strange attractors, localization, and symmetry breaking, etc. The main work can be summarized as follows:⒈Starting from the law of allmetric growth three fractal dimensions in a broad sense are derived, and according to these dimensions, geographical space is divided into three levels, i.e., real space, phase space, and order space, each of which corresponds to a kind of dimension.⒉Based on the idea of spatial disaggregation and using the RMI (relationship-mapping-reversion) principle, the urban system is formulated as three scaling laws of the three spaces, including number law, size law, and area law, which can be transformed into a set of power laws such as allometric law and Zipf's law associated with fractal structure. Thus scale-measurement-dimension constitutes the fundamental concepts of this project.⒊The scaling laws are derived from different perspectives, using entropy-maximizing methods in macro-level and utility-maximizing method in micro-level, consequently a pair of dual models based on the idea from nonlinear programming are built to interpret both the scaling laws and thus power laws. A discovery is that the energy and information can be transformed into each other in urban systems. The Braess paradox model is taken as an example to reveal the essence of entropy-maximization and utility-maximization which lies in the unity of the most efficient for the whole and the fairest for local parts and individuals.⒋The internal symmetry of the scaling laws are bring to light based on the dual model of entropy- and utility-maximization. The internal symmetry including translational symmetry, dilation symmetry, and bilateral symmetry originate from relative energy-conservation and information conservation of urban systems reflected by the dual models, and parity conservation of scaling laws. The external symmetry i.e. universality of geographical laws in urban sense is found to relate to the internal symmetry: the more symmetric a model is, the more universal the corresponding law is, and vice versa. This discovery is significant for us to select model and algorithmic.⒌The spatial order and the corresponding regularity emerge on the macro level of urban geographical systems because the symmetry is well-kept in macro sense. As for micro structure, the symmetry is broking and therefore we cannot find geographical laws in this level. The model should be aimed at macro level, but the parameters values are determined by micro activities. This paper includes a number of contributions on how to harmonize the macro and micro, global and local, random and structural, order and disorder, simple and complex, and so on, which is the significant task for the complexity exploration.⒍It is research course that is more important than others, but still several conclusions have been drawn as follows: regional speciality lies in spatial complexity, the process of spatial complication starts from symmetry rebuilding in macro-level while symmetry breaking in micro-level which accounts for emergence of complex regularity, and the target of geographical evolution seems to reconstruct the symmetry laws of nature.