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Hi,

Just to offer some update about Dense Julia patterns. 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.

Even cooler, these properties can be used to create periodic Julia patterns on the plane. The construction involves the geometrical concept of covering. The dense Julia sets live on the sphere, as I explained in a previous post. 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.

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