  In this paper, we introduce a couple of dynamical systems that are related to
the Chaos Game. We begin by discussing different methods of generating the
Sierpinski gasket. Then we show how the transition from random to uniform
selection reduces the Sierpinski gasket to simple periodic orbits. Next, we
provide a simple formula for the attractor of each of the introduced dynamical
systems based only on the contraction ratio and the regular n-gon on which the
game is played. Finally, we show how the basins of attraction of a particular
dynamical system can generate some novel motifs that can tile the plane.
