This post is a slight deviation from software to talk about the very interesting topic of Fractals…

Fractals mean different things to different people and are a very useful tool used in many ways, from tie-dye shirts to Star Wars animations to geographical measurements.

What are fractals?
Fractals are a mathematical construct, depicted as a geometric form that is composed of smaller pieces of itself. Thus it is possible to construct very complex shapes from very elementary ones using duplication.
A good article of fractals can be found at: http://mathworld.wolfram.com/Fractal.html

Benoit Mandelbrot invented the term  fractal, which he depicted using self-similarity and wrote a paper to compute the length of Britain’s coastline, which he depicted as a fractal.
The concept of self-similarity was not new and was around since the 1700s, but Mandelbrot defined it mathematically.
He defined a fractal dimension (figures composed of fractions) and faced opposition from classic mathematicians who were more about lines and curves and integral dimensions (whole parts made of ones and twos and so on).
http://math.bu.edu/DYSYS/chaos-game/node6.html#SECTION00060000000000000000 has a good explanation of this.

Mandelbrot defined a fractal to have the properties of self-similarity, fractal dimension and formation by iteration.What he essentially did was to describe nature in a mathematical form, which had never been done before when math was only applicable to human-made structures like buildings.

His paper was applied by many people to create complex shapes, e.g. artificial mountains were rendered by an animator at LucasFilms, who started with a simple triangle and replicated it multiple times. This started a whole world of realistic animations of natural forms like clouds, waves, etc.
Some practical applications are:
Cell-phone antennas where surface area is maximized by using a fractal shape.
Calculating the surface area of a forest to know how much CO2 it can absorb, which in a study was found to be fractal in nature (pun intended), and has the same form as that of a tree.
Similarly, clouds or oceans can be depicted as fractals and used for predictions.
A heartbeat waveform is apparently a fractal and not a clean sine or square wave. Thus irregular heartbeats can be detected if they don’t exhibit a fractal nature.

Here is a fractal I drew called the Pythagoras Tree, that starts with 1 triangle with 2 half-size triangles at 45 degrees, and that is repeated over. Hope this opens up the wonderful world of fractals for you…