It won grand prize in Sun Microsystem's Java programming contest in the Summer of 1995. All that students need to know to follow the proof is that triangles of equal height drawn on the same base have equal areas. The first applet is the one to use when introducing the topic, as it demonstates how the spirographs are made. The second one generates the final pattern must faster. The third extends the idea to a wheel travelling around a wheel travelling around a wheel.
Some fascinating patterns emerge.
This is not just a Java applet, it is an entire unit on fractal geometry with interactive applets included as needed. Very simple, and very effective. Then create a sampling distribution of the mean, or one of many other choices of summary statistics. This applet helps students understand confidence intervals. This applet demonstrates the central limit theorem using simulated dice-rolling experiments.