by Alistair Turnbull
The left-hand panel of the applet shows the Mandelbrot set. Click on it to choose a Julia set, which will be shown in the centre panel. Wave the mouse over the Julia set to choose a periodic cycle. There's a repelling periodic cycle near any point on the boundary of the set, and if the set has an interior then there's also exactly one attractive periodic cycle in the interior. (The applet won't find cycles with more than 20 points). Having chosen a (repelling) cycle, click again to draw the same Julia set linearized around it. The result should be a perfect geometric spiral. The white square shows a region that is exactly similar to the entire panel. The centre of enlargement and rotation is also marked.
This applet should work on Java versions 1.3 and later. Let me know if it works on earlier versions of Java too.