O espaço de Hausdorff e a dimensão fractal: estudo e abordagens no Ensino Fundamental

Abstract

Fractal geometry consists in the study of shapes established by simple or complex re cursive processes that take on high complexity for a sufficiently large number of iterations. So named in the 20th century by mathematician Benoit Mandelbrot, it presents peculiar properties, being the fractal dimension its main characteristic. In the case of fractals, their dimension assumes non-integer values, unlike the Euclidean and topological dimensions. This is due to the irregularity occupied by a fractal in the metric space where it is inserted. To calculate the fractal dimension, we use concepts of topology to characterize a complete metric space and verify its validity for the Hausdorff Space, in which we can calculate the Hausdorff dimension and use the Box-Counting method. Finally, we try to intuitively present this concept to students in the final years of elementary school through suggested activities that relate fractals to the mathematical skills of their study cycle, stimulating concrete measurement activities, constructions using concrete materials, investigations and algebraic generalizations.