F ractal D imensions

Abstract The term ‘fractal’ (from the Latinfractus, meaning ‘broken’), introduced by Benoit Mandelbrot about 25 years ago, is used to characterize spatial and/or temporal phenomena that are continuous but not differentiable. Geometrically, a fractal is a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced‐size copy of the whole. Fractal structures do not have a single length scale, while fractal processes (e.g. time series) cannot be characterized by a single time scale. Fractal theory offers methods for describing the inherent irregularity of natural objects. In fractal analysis, the Euclidean concept of ‘length’ is viewed as a process. This process is characterized by a constant parameterDknown as the fractal (or fractional) dimension. There are many fractal dimensions introduced in mathematical and physical literature. Fractal dimensions can be positive, negative, complex, fuzzy, multifractal, etc. The two most commonly used are the Hausdorf dimension and capacity.