Dense Julia sets on the sphere

[s31415]

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

[DarkBeam]

Question; can you give an example of the formula? And it escapes ... or not? Please

[s31415]

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.

[DarkBeam]

Interesting thank you.

[DarkBeam]

Here is a UF formula;

Code:

LGNabc {
; Sam's convergent Mandelbrot
; Use Pseudo magnitude as inner coloring
init:
  if @cu == 0
  z = @start
  c = #pixel
  else
  z = #pixel
  c = @start
  endif
  zold = 0
loop:
  z = z*z
  z = (@r1*z + @q1) / (@r2*z + c)
  zb = | zold-z |
bailout:
  (zb >= @cba) && (|z| < @bailout)

default:
  title = "Sam's convergent Mandel"
  complex param q1
    caption="c1"
    default=2
  endparam
  complex param r1
    caption="Rotate / zoom z up"
    default=1
  endparam
  complex param r2
    caption="Rotate / zoom z down"
    default=1
  endparam
  float param bailout
    caption="Bailout value"
    min=0
    default=1e20
  endparam
  float param cba
    caption="Bailout value (cvg)"
    min=0
    default=0
  endparam
  param start
    caption = "Seed/Perturb"
    default = (0,0)
  endparam
  param cu
    caption = "Current mode"
    enum = "Mandel" "Julia"
    default = 0
  endparam
  param sw
    caption = "Switch to"
    enum = "Mandel" "Julia"
    default = 1
  endparam
switch:
  type = "LGNabc"
  q1 = q1
  r1 = r1
  r2 = r2
  bailout = bailout
  cba = cba
  start = #pixel
  cu = sw
  sw = cu
}

Nice!

[s31415]

Yes, that's a Mandelbrot-like fractal. Now use the switch feature in some dusty region, and you'll get a dense Julia set.

[DarkBeam]

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

[s31415]

Actually, the Julia set is not quite dense, there are some voids remaining. Dense Julia sets really fill the plane with patterns, look at the ones in my gallery.

Also, here is a video showing the transition between non-dense and dense Julia sets:
<a href="https://vimeo.com/moogaloop.swf?clip_id=31325295&amp;server=vimeo.com&amp;fullscreen=1&amp;show_title=1&amp;show_byline=1&amp;show_portrait=0&amp;color=01AAEA" target="_blank">https://vimeo.com/moogaloop.swf?clip_id=31325295&amp;server=vimeo.com&amp;fullscreen=1&amp;show_title=1&amp;show_byline=1&amp;show_portrait=0&amp;color=01AAEA</a>

Finally, I forgott to mention that Dan Wills has been exploring these dense fractals even before me, check his blog
http://ultraiterator.blogspot.fr/

[Syntopia]

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.

[s31415]

Very nice!

[cKleinhuis]

nice formula, deforms smoothly! excellent!