Sand pendulums are an easy and beautiful method for students of all ages to learn a bit about the harmonic motion. These devices are easy to build and can be adjusted to give a variety of patterns. A pendulum that begins with two point of support can alter the periods of the pendulum as it moves on two perpendicular axis at the same time. The video gives simple instructions on the materials needed and how to adjust the configuration to vary the design.
Update: Yeany also shares how to make Lissajous patterns with light by using a small LED source, a dark room, and a camera with long exposure capabilities. He recommends leaving the shutter open “for times of about 15 seconds up to about 6 minutes.”
Further reading: Lissajous curves at Wolfram Mathworld, plus this interesting wikipedia note: “Not to be confused with spirographs, which are generally enclosed by a circular boundary, whereas Lissajous curves are enclosed by rectangular boundaries.”
And from Yeany: A home made string shooter & slow moving waves in rope.