This really is just the weirdest damn thing. It’s not the least bit clear to me why this would be true.
This morning I was reading Terry Tao’s overview of the work of Yves Meyer and ran across this line:
The powers φ, φ2, φ3, … of the golden ratio lie unexpectedly close to integers: for instance, φ11 = 199.005… is unusually close to 199.
I’d never heard that before, so I wrote a little code to see just how close golden powers are to integers.
Here’s a plot of the difference between φn and the nearest integer:
https://www.johndcook.com/blog/2017/03/22/golden-powers-are-nearly-integers/