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 degrees rotation and a root 2 zoom factor. We also observe invariance properties for the associated Dragon curves. Moreover, replacing square lattice paths by paths involving 2k pi/N-turns, we find analogous results using more general shuffles. (C) 2007 Elsevier B.V. All rights reserved.