One good example is a fern leaf. Each part is the roughly the same as the whole ie: break a leaf off of the original and it looks like the original – break a leaf off of that leaf and that looks like the original also. In other words the pattern of growth repeats over and over on ever decreasing scales here is an image of a fern leaf.

fern leaf

and below is a fractal fern leaf

fractal fern leaf

The fractal fern leaf, like the Mandelbrot set, is created using another iterated function system, this time using a simple (x, y) coordinate system. 20 iterations of the function f(x, y) are performed. f(x, y) for each iteration differing depending on the value of a random number.

The value (x’, y’) = f(x, y) is calculated using the following probability distribution:
Probability x’ y’
0.01 0 0.16y
0.85 0.85x + 0.04y -0.04x + 0.85y + 1.6
0.07 0.20x – 0.26y 0.23x + 0.22y + 1.6
0.07 -0.15x + 0.28y 0.26x + 0.24y + 0.44

After performing 20 iterations of the variable function f(x, y), the point, (x(20), y(20)) is plotted in green on a black background. (Here (x(n), y(n)) is the point after performing n iterations of the variable function f(x, y), starting with the initial point (x(0), y(0)). A new point is then chosen, and the process continues ad infinitum, until the user hits a key on the keyboard to stop processing. It doesn’t take very long at all to get a good “Impressionistic” fractal fern leaf, even on an old Intel Pentium or 486. The leaf just gets more “fleshed out” the longer the program is run.

To download software to create this fern fractal on your own pc visit the link below