This quite mesmerizing animation suggests there is a basic relationship between the properties of the circle, and perhaps more specifically to $\pi$, and the Fibonacci series and golden ratio.
Could someone explain what's going on here?
Source: Tumblr
I had a closer look and as pointed out in the comments, it looks cool but that's all there is to it. A still of the animation makes this painfully clear. The line is just drawn from a corner of the Fibonacci-related structure to the intersection with the circle. That is, it could have been any shape that rotated, and no relation to the circle's properties exists whatsoever.
They tricked me, I'll give them that, and I think I fell into trance a couple of times in the process...