A fractal is an image in which a pattern repeats itself infinitely many times.
You can construct some fractals by repeating the same geometrical operation over and over again; repeating the same procedure like this is called iterating. The geometrical operation which is repeated is called the iterated function. The fractal thus generated will have exactly the same pattern if you zoom in. You can generate these kind of fractals by using GeoGebra. By introducing random variables you can use these kind of fractals to generate pictures similar to nature. Random fractals are used to generate fractal landscapes.
Some types of self-repeating fractals, are easily generated by using turtle graphics and recursion. The example below is made using Scratch.
It is also possible to make 3D-fractals using turtle programming. Some examples are shown below.
Other fractals are generated by using a recursive formula on a point over and over again. Typically the point will either tend to infinity or not. When generating fractals like these you measure how long it takes for a point to "escape" outside some given boundary; they are therefore called Escape-time fractals. These fractals do not have exactly the same pattern when you zoom in, but the structure of the pattern is the same.
Fractals can also appear as strange attractors to nonlinear dynamic systems. Nonlinear dynamic systems can exhibit unpredictable behaviour due to the fact that the error grows exponentially as you iterate. Even though you cannot predict exactly where a point will end up, you can predict that it will end up at the so called strange attractor, an attractor that has a fractal structure. Note that there is nothing random in the system itself, the behaviour is seemingly chaotic but it is deterministic; it is thus called deterministic chaos.
An interactive visualisation of the Lorenz attractor in 3D is shown at Interactive Lorenz Attractor.
When using the Newton method on complex numbers, the complex plane is divided into regions separated by the so called Newton fractal. In the example below, the complex plane is divided into three regions (red, green, blue).
Benoit Mandelbrot: Fractals and the art of roughness ( on TED)
by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License