An Approximative Calculation of the Fractal Structure in Self-Similar Tilings

Fractal nature is observed in our real-life infrastructures of urban spatial organization and many technological networks. Indeed, the fractal dimensions of landuse and area-perimeter have been measured in real data of urban cities [1][2]. Similar structures with mixing of dense and sparse areas of nodes have been found in router networks [3], air transportation networks [4], and mobile communication networks [5]. These studies show that the spatial distribution of human activities is inhomogeneous and concerning with the population density. Thus, there commonly exists a hierarchical structure based on statistical self-similarity beyond regular mathematical models such as Sierpinski gasket and carpet. In other words, fractal behavior is not limited to objects with a regular morphology, but can be introduced by iterative evolutionary processes of subdivision or growing at different levels of scaling.