Persistence intervals of fractals

Objects and structures presenting fractal like behavior are abundant in the world surrounding us. Fractal theory provides a great deal of tools for the analysis of the scaling properties of these objects. We would like to contribute to the field by analyzing and applying a particular case of the theory behind the P.H. dimension, a concept introduced by MacPherson and Schweinhart, to seek an intuitive explanation for the relation of this dimension and the fractality of certain objects. The approach is based on recently elaborated computational topology methods and it proves to be very useful for investigating scaling hidden in dimensions lower than the “native” dimension in which the investigated object is embedded. We demonstrate the applicability of the method with two examples: the Sierpinski gasket–a traditional fractal–and a two dimensional object composed of short segments arranged according to a circular structure.