I have been studying discrete dynamical systems for a few years while simultaneously applying the theory to the creation of generative art. I have recently been focused on the IFS (Iterated Function Systems) in the plane whose functions are two similitudes (conformal affine maps). Making generative art strictly in the context of these simple order two IFS has played a large role in leading me to the parts of the subject I find most aesthetic, elegant, and interesting. The images I have chosen for the Bridges Conference are cross-sections of the 'Connectedness Locus,' of the IFS of two similitudes. The connected locus exists as a subset of the parameter space of the IFS of two simlitudes, and represents the IFS with connected attractors.
This image was generated in a similar way to the previous one, however it is in a cross-section which is defined instead by those pairs of similitudes whose linear components differ by a rotation through 60 degrees. The subset of the connectedness locus in this cross-section has bilateral planar symmetry. This image centers on one of the two intersections of the boundary of the subset of the connectedness locus with the bilateral axis of symmetry. The set here is seen at a scale of 10^-8.
This image of the connectedness locus is from a cross section defined by those pairs of similitudes whose linear components differ from each other by a rotation through 90 degrees. The set has been translated to center exactly the special location where this waves crest is curling into a point. The wave's are reminiscent of those that appear in the famous Mandelbrot set, which is also a connectedness locus; although I find the IFS of two similitudes to be much less delicate than the Julia sets whence the Mandelbrot set is defined, and hence the features of the Mandelbrot set itself are likewise more spindley, than the more robust structures we see above.