Applications of Fractal Analysis in Science, Technology, and Art: A Case Study on Geography of Ukraine

Abstract

Fractal sets have been known for more than a century. However, only in 1975, Mandelbrot gave them the name “fractal” and mathematically defined them as sets whose Hausdorff dimension exceeds the topological dimension. While initially fractals were a pure mathematical phenomenon, afterwards fractal-like properties have been found in many natural and artificial objects and processes. Fractal theory and fractal analysis have been developed and applied in many different areas, including biology, health care, environmental studies, urban planning, geology, geography, chemistry, ecology, astronomy, computer science, social science, music, literature, art, and so on. This paper gives a survey of some of the applications of the fractal theory and analysis to solve different problems in various areas. We give basic definitions and describe frequently used algorithms to compute a fractal dimension. Finally, we apply the fractal analysis to study the geography of Kyiv and Ukraine.