Fractals Fractals Fractals Fractals are defined as self-similar figures. That means that they appear exactly the same no matter how what the zoom. The word Fractal comes from frangere, to break, and fractus which means irregular and fragmented.
Most fractals also have an infinite perimeter and a finite area.
This is simply demonstrated by a Koch Snowflake. It is easy to see that the perimeter is infinitely long by the area never exceeds that of the circle in which the base triangle was inscribed. Now for the math. Dealing with these complex numbers is easier then it looks. Here's a nice short one.