On different forms of self similarity

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. (C) 2016 Elsevier Ltd. All rights reserved.