Novel Planar Self-Similar Fractal Sets with a Variable Hausdorf Dimension

Abstract

Two different approaches to generalizations of the classical two-dimensional Sierpinski carpet are presented in correct mathematical sense. The first generalization is based on classical principle of fractal carpet creation. It relates to the change of the fractal creator. In the result a series of two-dimensional self similar fractal square sets can be arisen. The second generalization is reworked on the base of one-dimensional self similar fractal sets with a variable fractal dimension. For all generalized classes of self similar fractal square sets have been presented basic elements of their mathematical analyses.