Fractals aspects in urban structures.— The introduction of fractals by B. Mandelbrot in the scientific discussion has given rise to a real boom of applications in all possible domains. In the present paper their application to urban structures will be discussed. The investigation of different aspects of the urban patterns, like the spatial distribution of the built-up areas or of different types of exploitation (supply systems, administration, etc.), as well as the dendrification of transportation networks leads to the more general point of view, that fractal dimensions are a useful quantitative mesure for all types of hierarchically organized subsystems. This can serve to compare those subsystems for different urban units. Finally, fractal growth processes, their simulation, and their growth dimension are introduced with respect to urban growth.

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Fractal geometry can be used for determining the morphological boundaries of metropolitan areas. A two-step method is proposed here: (1) Minkowski's dilation is applied to detect any multiscale spatial discontinuity and (2) a distance threshold is located on the dilation curve corresponding to a major change in its behavior. We therefore measure the maximum curvature of the dilation curve. The method is tested on theoretical urban patterns and on several European cities to identify their morphological boundaries and to track boundary changes over space and time. Results obtained show that cities characterized by comparable global densities may exhibit different distance thresholds. The less the distances separating buildings differ between an urban agglomeration and its surrounding built landscape, the greater the distance threshold. The fewer the buildings that are connected across scales, the greater the distance threshold.