Fractal Geometry: Historical and Conceptual Background

Fifty years ago, the term fractal was introduced by Mandelbrot in his book ’Les objetfractals: forme, hasard et dimension’. Fractal geometry deals with intricate patterns andirregular shapes possessing the scale and/or conformal symmetry. Presently, the fractalgeometry serves as a framework for studies of complex systems of diverse nature. One ofthe most fundamental geometric conceptions is the concept of symmetry. Differentgeometries can be classified according to the group of transformations under which theirpropositions remain true. In particular, the key symmetry of the fractal geometry is thescale invariance. Another crucial paradigm in the fractal geometry is that differentproperties of a fractal pattern are governed by different dimension numbers, at least one ofwhich differs from the topological dimension. Accordingly, the inherent features of afractal pattern are characterized by a set of generally independent dimension numbers. These numbers allow for the classification of fractal patterns. In this review we brieflysurvey the historical background and the conceptual foundations of fractal geometry.