Title: Periodic Julia patterns Post by: s31415 on March 31, 2012, 12:02:29 AM Hi, Just to offer some update about Dense Julia patterns (http://algorithmic-worlds.net/blog/blog.php?Post=20120316). It turns out they have very cool invarance properties under conformal transformations. Under such a transformation, while the details of the pattern change, its general look stays invariant. Some examples in pictures can be found here (http://algorithmic-worlds.net/blog/blog.php?Post=20120325). Even cooler, these properties can be used to create periodic Julia patterns on the plane. The construction involves the geometrical concept of covering (http://en.wikipedia.org/wiki/Covering_map). The dense Julia sets live on the sphere, as I explained in a previous post (http://www.fractalforums.com/mandelbrot-and-julia-set/dense-julia-sets-on-the-sphere/). On the other hand, a periodic pattern can be seen as a decoration of a torus. The existence of a covering of the sphere by a torus allows to pull-back the Julia pattern from the sphere conformally and create a periodic pattern. More details and pictures can be found in this blog post (http://algorithmic-worlds.net/blog/blog.php?Post=20120330). This last post also exemplifies why non conformal transformations are aesthetically much less interesting than conformal ones, as far as Mandelbrot/Julia-type fractals are concerned. Attached are two pictures of dense Julia patterns. Check the blog post for higher resolution pictures. Sam |