Introducing the magic cycle button
The "cycle" button is an attempt to show the effect of the z->z2+c transformation.
Even newer, even more improved applet
There's nothing hugely exciting going on here. One notable detail is that it isn't enough to apply the transformation to the pixels in th bitmap; that would result in terrible accuracy and hideous scaling effects. I modified the code to remember the exact value of each number plotted and use those bvalues in the transformation.
I considered a few ways to interpolate but eventually settled for a very simple solution: linear interpolation. Each point just moves in a straight line from its starting position to its image under the transformation. This has the advantage of keeping the two fixed points stationary. (It also is angle-preserving, but I probably wouldn't consider anything that wasn't).