I am a fan of the Mandelbrot set since I have seen it the first time. I developed a screen saver for windows which displays the Mandelbrot set and zooms in randomly.
I also tried to develop something similar for the iPhone but did not succeed for the time. When I was investigating possibilities to draw lines in HTML code, I saw that with HTML 5 there is a new tag <Canvas> which allows exactly what I need.
Here is the result. I have created a page which shows the Mandelbrot set and the same for the Julia set. Both are very impressive. You can zoom in by clicking somewhere in the graph and even move the graph with the mouse.