END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

Hi,

I've been exploring for a while "dense" Julia sets, i.e. Julia sets filling the whole plane. But actually, it turns out that they can be more naturally pictured on the sphere! A few of these Julia sets can be seen as full panoramas with the help of a cool Java applet. Check this collection of pictures:
http://algorithmic-worlds.net/expo/expo.php?Collection=Spherical&CollSearch=0
Once you click on a thumbnail, look for the link at the bottom of the page to see the full spherical Julia set.

More explanations can be found here:
http://algorithmic-worlds.net/blog/blog.php?Post=20120316

Sam

Yes, I tried to explain it in the blog post, but maybe it wasn't so clear. Just iterate the following:
Take the square:

z -> z^2

then instead of simply adding a constant, perform a Moebius transformation:

z -> (az+b)/(cz+d)

Effectively this amounts to

z -> (az^2+b)/(cz^2+d)

Take for instance a = 1, d = 1 and explore a bit with b and c. For some values of the parameters, the Julia sets fills the whole plane (or sphere).

To find the values of the parmeters yielding dense Julia sets more easily, you can also draw a Mandelbrot-like fractal, by making either b or c to be equal to the value of the pixel.  Values associated to dense Julia sets appear in the form of "dusty" regions.

To answer your second question, the orbits do not escape for dense Julia sets. If an orbit escapes, it means that it converge to infinity, so that the corresponding point is "smooth" and that it does not belong to the Julia set. To color dense Julia sets, I use essentially the mean distance of the orbit to the origin, with a few fancy tricks. I explained this a bit here, in the case of Ducks fractals:
http://algorithmic-worlds.net/blog/blog.php?Post=20110319

To generalize further the formula above, just take the ratio of any two polynomials in z. If you tune the parameters, you'll find dense Julia sets as well.

Here is a UF formula;


Nice!

More misty Mandelbrot images; last one is a "dense" Julia (well probably)

Funny, some weeks ago I wondered if spherical fractals existed :-)

Here is Fragmentarium image where the fractal plane is depicted together with the stereographical projection on the Riemann sphere. It is a nice way to be convinced these fractals fit naturally on the sphere. The script will be part of the next release.

nice formula, deforms smoothly! excellent!