Here a line of fixed length is moved along the edge of an ellipse, tracing out a collection of new shapes. Consider the area of the shape traced by a point p units from one end of the line and q from the other. Holditch’s theorem, regarded as a milestone in the history of maths, tells us this area is less than the area of the ellipse by at least π×p×q. Curiously, this formula holds not just for an ellipse, but any closed curve. [more] [code]
Bouncing balls in a circle gives one of the simplest systems to exhibit chaos, as was pointed out in a comment by Andrew Moylan. The animation above shows two balls which start off with almost exactly the same speed and location, but before long they are travelling along completely different trajectories. Such high sensitivity to the initial conditions defines chaos.