Fractals

Julia sets and Mandelbrot sets

Abstract

The fractals were named in the early 1980’s by B. Mandelbrot, to classify certain objects that have no integer dimension (1, 2, ...), but fractional. Fractals are figures too irregular to be described in the language of traditional Euclidean geometry. Different definitions of fractals emerged with the improvement of his theory. Without mathematical rigor fractals can be defined as objects that exhibit self-similarity, that is, a fractal is an object whose geometry repeats itself endlessly into smaller portions, similar to the object itself. There are several types of fractals, but we present the figures generated from iterations of functions. But to generate the fractals figures, we need to use iterations of complex functions, that associate a complex point a + bi

to a complex image f(a+bi) = c+di. The Julia set is known as the set that separates the complex plane into two sets, the first is formed by the points whose orbits tend to the origin and the second by the points whose orbits tend to point at infinity. The points of the Mandelbrot set provide us connected Julia sets and those points that are not in the Mandelbrot set correspond to unconnected Julia sets. Julia and Mandelbrot sets are of fractal geometry and in this article are discussed the dynamics of complex quadratic function f(z) = z2 + c.