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”.)

## About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.

YES! Nice work. Its like a zero mass, zero gravity triple pendulum. 😉

And you’re right (of course) this is not a hypotrochoid, even without the 3rd term. And not just because orange is going the wrong speed for it, but its the wrong direction too – I hadn’t noticed that before.

I do like the simplicity of the exponential form, but I wonder if commutativity complicates the analysis a little – if you think of each rotation as relative to the parent, their rotation speeds are 1,5,-20, right? To me it seems a little easier to understand the factors that way…

Haha, yes, a zero mass, zero gravity triple pendulum is a nice way to think about it. =)

Thinking of the speeds relative to their parents is an interesting approach. I don’t think commutativity actually complicates the analysis at all, because you are always just taking pairwise differences, and the order has no effect on those (up to differences in sign).

How about a 7 or a 41 segment pendulum? There was a related post on hacker news the other day, and I thought of you…

https://news.ycombinator.com/item?id=9762022

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Very nice! I love how so many of us have spent time playing with this curve. Big thanks to John D Cook for bringing it to our attention.