Here's an animation I made with Flash. For every one trip back and forth that the top circle makes, the second circle makes 2 trips, the third 3 trips, and so on. Try with 2 or 3 objects to see better what's happening, then explore larger numbers (100-300 is nice).
Despite the simplicity of the math, there are lots of crazy patterns that pop up as you watch the strands weave and unweave. There are connections here to factors and primes. Notice how you can catch sight of, for example, 4 strands, then 3, then 5, then 2, then ...? What is happening? What is the pattern?