Geometria fractal: Propriedades e características de fractais ideais

Abstract

Recent discoveries reveal that mathematical models, established a long time ago and searching to reproduce the nature's geometry, sometimes result being incomplete and even inadequate in some situations. Specifically, many of the forms found in the nature are not circles, triangles, spheres, icosahedrons or rectangles. Finally, they are not simple curves, surfaces or solids, as defined in the classical geometry of Euclides (300 b.C), whose theorems possesses a prominent place in the geometry texts. In this work a brief and elementary, although intended to be cosistent, discussion about some definitions and applications related to the fractal geometry is presented. It is also presented properties of some fractals that, for its historical importance or wealth of characteristics, constitute "classical" illustrative examples of the fractals properties which, despite this, many times appear dispersed in the specialized literature. It is shown, by construction, that the measures of length, area and volume for these objects, within the usual Euclidean dimensions, lead to contradictory results. This can be explained by considering that these objects can be adequately measure using spaces of fractional dimensions.