In this paper we study the DEM (Distance estimation method) algorithm, which enables effective graphical representation of fractal sets. The algorithm is based on the theory of complex dynamics, which studies complex function's behavior under repeated iterations. This branch of mathematics had its beginnings between the years of 1917 and 1919, when the first research about iteration of complex rational function of one variable was published by French mathematicians Gaston Julia and Pierre Fatou. There followed multiple years of inactivity, until it was disrupted by the progress in the field of computer science and hence fractal geometry. Thus the field became immensely popular amidst mathematicians and artists alike. In the midst of this paper we will study one of the most recognizable objects of the theory, the so called Mandelbrot set. The name belongs to a subset of the complex plane, which in its own way illustrates the family of quadratic polynomials with a connected Julia set.
We will define this set and prove some of its topological characteristics. The main proof being its connectedness, which will allow us to derive the DEM algorithm in the last section, using which we will be able to effectively represent the set graphically.
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