Mandelbrot fractals and incomplete similarity

The concept of fractals. Fractal curves

In the scientific and even popular literature of recent time fractals have been widely used and discussed. By fractals are meant those geometric objects, curves, surfaces and three- and higher-dimensional bodies, having a rugged form and possessing certain special properties of homogeneity and selfsimilarity. Such geometric objects were studied intensively by mathematicians at the end of the nineteenth century and the beginning of the twentieth century, euphony particularly in connection with the construction of examples of continuous nowhere-differentiable functions. To many pure mathematicians (starting with Hermite) and most physicists and engineers they seemed for a long time mathematical monsters having no applications in the problems of natural science and technology. In fact, it is not so and in clarifying this point the concept of intermediate asymptotics plays a decisive role.

The revival of interest in such objects and the recognition of their fundamental role in natural science and engineering is due primarily to a series of papers by Mandelbrot and, especially, to his monographs (1975, 1977, 1982). Mandelbrot coined the very term ‘fractal’ and introduced the general concept of fractality.