Non-linear iterated function systems and the creation of fractal patterns over regular polygons

We describe how non-linear iterated function systems allow one to create fractal patterns over regular polygons and over disks. The iterated function systems are defined by four functions. Two of these are linear and the other two contain the square root of a complex variable. The parameters in these functions can be varied in an understandable way in order to explore the resulting fractals. The range of geometric art that can be created is further widened by application of the concepts of sth order restriction and domain restriction.