On different forms of self similarity

Abstract

Fractal geometry is mainly based on the idea of self-similar forms. To be self-similar, a shape must able to be divided into parts that are smaller copies, which are more or less similar to the whole. There are different forms of self similarity in nature and mathematics. In this paper, some of the topological properties of super self similar sets are discussed. It is proved that in a complete metric space with two or more elements, the set of all non super self similar sets are dense in the set of all non-empty compact sub sets. It is also proved that the product of self similar sets are super self similar in product metric spaces and that the super self similarity is preserved under isometry. A characterization of super self similar sets using contracting sub self similarity is also presented. Some relevant counterexamples are provided. The concepts of exact super and sub self similarity are introduced and a necessary and sufficient condition for a set to be exact super self similar in terms of condensation iterated function systems (Condensation IFS’s) is obtained. A method to generate exact sub self similar sets using condensation IFS’s and the denseness of exact super self similar sets are also discussed.

Introduction

Fractals and self similarity are two intense areas of current mathematical research. Fractals allow us to see the world as it is. Roughly fractal theory is the study of shapes of natural objects. Self similarity is a significant property of fractals. There are different forms of self similarity in mathematics and nature. They include super, sub, partial and quasi self similar forms.

Fractals were introduced and studied by Mandelbrot [3] for the first time in 1975. Then many mathematicians including Barnsley[15] and Falconer [6], [7], [8], [9], [10], [11], [12], [13] carried the research forward. Self similarity was mathematically defined by Hutchinson [5] in 1981 and the concepts of sub and super self similarities were introduced by Falconer[9] in 1995 and subsequently studied by Bandt [4], Schief [1], [2], Mcclure and Vallin[16] and many others. More related works can be seen in [17], [18], [19], [21], [23].

Abstract spaces are abundant in mathematics, physics and other areas of science and technology. Even though it is impossible to visualize functions, surfaces and different types of sets in these spaces, they do exist in some complicated mathematical form. For example, in physics a ‘state space’ is an abstract space in which different positions represent different states of the physical system. In quantum mechanics theory, a state space is a complex Hilbert space. Also, the four dimensional ‘space time’ may have some fractal properties on a quantum scale. Fractals exist in abstract Sobolov spaces as self similar energy forms [20]. It is possible that self similar fractals exist even in the sub atomic levels.

Even though the complete structure of fractals in abstract spaces are difficult to obtain, some of their properties like fractal dimension can be studied and conclusions can be obtained. It is shown in [14] that the typical trajectories of quantum mechanical particles are continuous but not differentiable. But they can be characterized by their fractal dimension.

Having these in mind, some of the existing results in ℜⁿ (Corollary 2.2[16] and Theorem 3.3[16]) are generalized to arbitrary spaces. Also, for the mathematical completion, our definitions will be given in abstract metric spaces. But all the examples will be in ℜⁿ, particularly in ℜ², for a better understanding.

Even though all forms of self similarity are present in nature, super self similarity is the most common and abundant one. Most of the objects in nature like trees, ferns, flowers etc., are super self similar. In this paper, we look super self similar sets more closely and discuss some of their properties. Also, some new forms of self similarity are introduced and related results are discussed.