Spirography

Suppose you had a circle made of a hundred or so equidistant points on its circumference. If you drew straight lines connecting distant points on this circle (chords) in some systematic way, you’d produce a kind of spirograph. I don’t know what these plots are actually called, and “spirograph” is actually just the brand-name of a popular toy, but in any case, most people have probably drawn something like this at some point in their lives.

The most common form of these that I’ve encountered are ones where the pattern for connecting the points is to simply connect every point p_i to p_{i+k} where k is some fixed offset. What if instead we connected p_i to p_{ik} where k is now a fixed multiplier?

I used the Desmos graphing calculator to play around with different values of k in real time. I did not expect the “lobe” count to scale as \propto k-1.

I wonder if someone could come up with some hand-wavy argument as to why this behaves this way? In any case, you can try it out at Desmos here.

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