As a follow-on to my previous post, here’s an animation (17MB) showing how the “mystery curve” arises as a sum of circular motions:
Recall that the equation for the curve is . The big blue circle corresponds to the term—it is a circle of radius and makes one complete revolution before the animation restarts. The medium orange circle corresponds to : it has a radius of and rotates times as fast as the blue circle. The small green circle corresponds to . It rotates times as fast as the blue circle, but in the opposite direction. It has a radius of , and starts out rotation out of phase with the others, since multiplying by corresponds to a rotation in the complex plane. (Notice that whenever the blue and orange circles are pointing in exactly the same direction, the green circle is perpendicular to them.)
One interesting thing to note is that addition of complex numbers is commutative—which means that we could just as well put the fast green circle in the middle, and the big blue circle next, and the orange circle on the outside, or any other ordering. The red point would always trace exactly the same curve.
Edited to add: can you see how the numbers 1, 6, and -14 result in 5-fold symmetry? Hint: how many times per day do the hands of a clock line up? (Farris proves this analytically in his book, but it wasn’t until staring at this animation that I feel like I really “got it”.)