The feathery sections look a lot like images generated by iterated function systems (IFS). I wonder if the roots of some polynomial using parameters of the IFS generator as coefficients are exactly those points that will be drawn by the IFS generator, hence the similar patterns. Seeing the random-looking patterns gradually converge reminds me of playing with IFS parameters until patterns start to form and gradually coalesce into a solid shape.
http://news.ycombinator.com/item?id=1033455
http://news.ycombinator.com/item?id=980043 <- This has some discussion