Fractal Dimensionalities

Abstract

One of the most important characteristics of a fractal is its dimensionality. In general, there are several options for mathematical definition of this value, but usually under the object dimensionality is understood the degree of space filling by it. It is necessary to distinguish the dimensionality of space and the dimension of multitude. Segment, square and cube are objects with dimensionality 1, 2 and 3, which can be in respective spaces: on a straight line, plane or in a 3D space. Fractals can have a fractional dimensionality. By definition, proposed by Bernois Mandelbrot, this fractional dimensionality should be less than the fractal’s topological dimension. Abram Samoilovich Bezikovich (1891–1970) was the author of first mathematical conclusions based on Felix Hausdorff (1868–1942) arguments and allowing determine the fractional dimensionality of multitudes. Bezikovich – Hausdorff dimensionality is determined through the multitude covering by unity elements. In practice, it is more convenient to use Minkowsky dimensionality for determining the fractional dimensionalities of fractals. There are also numerical methods for Minkowsky dimensionality calculation. In this study various approaches for fractional dimensionality determining are tested, dimensionalities of new fractals are defined. A broader view on the concept of dimensionality is proposed, its dependence on fractal parameters and interpretation of fractal sets’ structure are determined. An attempt for generalization of experimental dependences and determination of general regularities for fractals structure influence on their dimensionality is realized. For visualization of three-dimensional geometrical constructions, and plain evidence of empirical hypotheses were used computer models developed in the software for three-dimensional modeling (COMPASS, Inventor and SolidWorks), calculations were carried out in mathematical packages such as Wolfram Mathematica.