Printable instructions for constructing a pentagon inscribed in a given circle.
Draw a circle on a piece of paper, and a random point inside. If you continually fold points on the edge of the circle on top of the point inside, then the fold marks will combine to form the shape of an ellipse. [code] [more] [inspiration] [bonus]
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.