Aplicação do método box counting para a estimativa da dimensão fractal de figuras planas digitalizadas

Abstract

Many forms in the nature can not be explained in the molds of the conventional mathematics. In order to fulfill this need scientists have developed the fractal geometry. The science of Fractals, since its origin, is the reason for several questionings. After all, how can we admit that certain objects have a dimension which is not entire? The Fractals were nominated instead of discovered or invented in the beginning of the eighties by Benoit Mandelbrot, to classify certain intricate objects which do not have a complete but fractional dimension. The Fractal Dimension can characterize a group or an object, for the first, it is the number that informs us how densely the group occupies the metric space where it is, and for the second, the irregularity of its contour. Determining the fractal dimension means to measure the complexity of objects. Different definitions of Fractals appeared with the improvement of its theory. The Fractals created a new segment of the mathematics, often designated as the geometry of the nature. The strange and chaotic forms of the Fractals describe some natural phenomenon, as the seismic, the development of the trees, the structure of its rind, these phenomenon have as a common characteristic its spontaneous generation. In this work, it was developed a semiautomatic system to esteem the fractal dimension of plane illustrations using the method called Box Counting for Fractal Dimensions. The developed method was applied to geometric illustrations, fractals illustrations as well as to rings of growth of forest species. Based on the experimental results and after making a discussion giving emphasis in the principal characteristics of the respective method.