I don't know, for me the fact that this gives you a visualization of a totient is really interesting.

Another really interesting idea is to deliberately change the thing which you are rationally approximating; you don't have to rationally approximate π if you don't want to, that's just if you make steps of 1 radian. Make steps of q radians and you get the denominators for rational approximations of q/π.

This is used in the golden spiral algorithm[1] to evenly-ish distribute points on a sphere, we choose the most irrational number q/π = φ, the golden ratio. Since all of its rational approximations suck, the spirals are as inoffensive as they can be.

1. https://stackoverflow.com/questions/9600801/evenly-distribut...