Fractals are mathematical entites which are characterized by a property of self-similarity on all scales. They are a special case of a more general class of systems called discrete dynamical systems, irregular and unpredictable enough to go unexplained in the light of Euclidean geometry.

Dynamical systems are objects of study ofcomplex analysis and complex geometry. Examples of such systems are those described by the logistic function z → kz(1-z) and by the complex function z → z²+c, where k is usually between 1 and 4 and c is a constant.

Benoit Mandelbrot was the first to note that fractals had fractional dimension. The coast line, for example, is not exactly one-dimensional and surely it is not two-dimensional. It is something inbetween. To determine the dimension of a certain fractal, let us consider a d-dimensional object divided in N equal parts. Moreover, let us consider that each of these parts is reduced to a scale r. Then the fractal dimension is given by d=ln(N)/ln(1/r). However, for fractals which do not exhibit exact self-similarity, there are approximation methods to estimate their dimension. For example, the Sierpinski triangle can be divided into three equal parts, each reduced to 1/2 the original size. Then the fractal dimension of the Siepinski triangle is equal to d=ln(3)/ln(2)=1.58496.

Curiously, in nature some property of self-similarity tends to be present almost everywhere. If we look around us, we can see fractals or approximations thereof in ferns, snowflakes, tree branches and coastlines.

The main types of fractals we can identify are:
Iterated Function Systems (IFS)
Lindenmayer Systems
Julia and Mandelbrot Sets