The tip of a robot arm moves in a horizontal plane, modelled as the complex plane with the pivot of the arm at the origin $O$ and distances in metres. The tip starts at $z_0 = 2 \sqrt{3} + 2 i$. A turn command rotates the arm anticlockwise about $O$ through $\frac{\pi}{6}$ without changing its length, so after $n$ turn commands the tip is at $z_n = \left(\mathrm{cis}\left(\frac{\pi}{6}\right)\right)^{n} z_0$.