Construções de fractais com o GeoGebra e dimensão fractal

Abstract

In this work, we will present the elements and concepts related to Fractal Geometry such as its definition, classification into types, properties, and some measurable characteristics such as area, perimeter, and dimension measurements. At first, we will focus on the characterization of the most classic fractals of the theory, such as the Sierpinski triangle, the Koch curve, the Cantor set, among others. We will show how to build these objects using the mathematical tool of geometric transformations and their matrix translations and the implementation of these concepts through GeoGebra dynamic geometry software, which allows us to build the vast majority of fractals that will be mentioned during the work. Finally, we will also present a study on the concept of fractal dimension, whose applications are vast in various areas such as Economics, Medicine, Biology, among others. More precisely, we will present two methods of obtaining a fractal dimension, the first using the Hausdorff-Besicovitch method and a second way using the box-counting method. We believe that this monograph can be used as the first guiding material for studies and research in the field of Fractal Geometry, mainly due to the richness of details, especially for those who are unaware or know little about the theory. In addition, by bringing construction methods with GeoGebra, we believe that the material also serves as a guide to guide the use of theory in the classroom for students of the Licentiate Degree in Mathematics, future teachers. Finally, for those who already have a basic knowledge of fractals, the dimension study serves as a basis for guiding the application of this object.