Shuffle operations on discrete paths

We consider the shuffle operation on paths and study some parameters. In the case of square lattices, shuffling with a particular periodic word (of period 2) corresponding to paperfoldings reveals some characteristic properties: closed paths remain closed; the area and perimeter double; the center of gravity moves under a 45∘ rotation and a zoom factor. We also observe invariance properties for the associated Dragon curves. Moreover, replacing square lattice paths by paths involving 2kπ/N-turns, we find analogous results using more general shuffles.