Thanks for the link; that's a great site.
Way, way back when I did A level physics I learnt about lissajous figures. We had a double pendulum (imagine a kid's swing with another smaller one attached below it but swinging at 90 degrees to the first. Below pendulum two was a leaky sandbag so that as each pendulum was set swinging the resultant trace of the trail of sand was a lissajous figure.
I thought that was quite fun. But then the dark ages passed and they invented computers. And I wrote a 10 line BASIC program to make the figures with the two rates of swing as independent variables. It was for a competition where you had to make a program that did something useful in 10 lines. (I didn't win because I was the teacher, setting up the task, and it was my demo)
Basically you just need two sine curves, one for the x coordinate and a different one for the y coordinate.