Fern

This is another example of an IFS fractal. As you can see each leaf of the fern is in fact the same as the full fern. This infinite level of self similarity is what gives this image its fractal dimension. Apart from the Sierpinski Gasket, the fern is probably the most well known of this style of fractal.

Sierpinski triangle (default)

This is a Sierpinski Gasket drawn using an Iterated Function System. This fractal was originally thought up by W. Sierpinski and predates the Mandelbrot Set. It was originally produced by starting with a triangle, cutting out the middle, and repeating the process infinitely.

In this way, you can see that at each iteration, one quater of the original triangle is removed. That is, three quaters of the area of the original triangle is left after the first iteration. From this observation, it is not hard to infer that after n iterations, the area of the gasket would be (0.75)ⁿ times the area of the original triangle. So after an infinite number of iterations, you would find there was no area at all.

Gaston Julia studied the iteration of polynomials and rational functions in the early twentieth century. If f(x) is a function, various behaviors can arise when f is iterated. Let's take, for example, the function f(x) = x² – 0.75.