Fractals: objects to better explain our world

Abstract

[en] In this paper we will look at some properties fractals and show the usefulness of one of them, the fractal dimension, for the study of natural and artificial phenomena. We will center our attention on fractals generated by Iterated Function Sets (IFS), which we will define as a system of contractive mappings on non-empty compact sets in a complete metric space. First, we will present some simple, well known, fractals, and show how to generate them with a geometrical construction. To compute these objects, we will use two distinct algorithms based on the iteration of IFS, a deterministic and a random one. We will then see an application of the fixed point theorem for IFS, named the Collage Theorem. We will show that an IFS’s attractor is unique and independent of the initial set. Moreover, we will show that both the deterministic and the random algorithms converge to the same limit: the attractor of the system. Having studied fractals generated by IFS, we will go on to look at fractal dimensions. For this purpose we will review the classic concept of dimensions, and broaden it to include non-integer dimension. We will see different types of fractal dimensions, some of which are suited to a specific type of fractals, such as the self-similarity dimension applicable to self-similar shapes, and a more general dimension, applicable to any fractal, namely, the Hausdorff-Besicovitch Dimension. We will also see the Box-counting algorithm, which approximates the Hausdorff-Besicovitch Dimension and is often used in its stead because of the complexity of calculating the dimension. We will conclude our exploration of fractal dimensions with the presentation of a small personal contribution to this area – our own version of a program which implements the box-counting algorithm on images. In the last chapter we will see some examples of the practical uses of the fractal dimension in fields of study as diverse as medicine, market research, image classification and so on. Through these examples we can appreciate the impact of fractals on our way of modeling or explaining our world.