Buddhabrot

[tit_toinou]

A Buddhabrot : http://titoinou.deviantart.com/art/Buddhabrot-100k-600k-286331491
With the technique http://erleuchtet.org/2010/07/ridiculously-large-buddhabrot.html.

[tit_toinou]

And a new Buddhabrot i juste made :

[ker2x]

Shiny \o/ 

[Eric Bazan]

I like the last, most colorful one the most. However they do not appear completely developed as there's an asymmetry about the real axis.

How are you coloring these?

[kram1032]

Those are awesome
So you ONLY take those orbits close to the border of the set?
Also, it LOOKS like you colour each and every orbit you find in a different random colour?
Really nice stuff

Eric: If you look at the links he provided, you'll see that the asymmetry is basically wanted (but a result of the technique sampling only a limited range of escape times and thus dropping out a lot of stuff)
Though if you really want symmetry, you could always just mirror it and do an overlay...

[Dinkydau]

The last one looks really awesome!

[aluminumstudios]

That second one is really outstanding.  The coloring is great and its so smooth looking.  Excellent work!
 

[tit_toinou]

Thanks for the comments.

@Eric Bazan: The asymmetry is wanted for aesthetics purposes. And you're right, theses Buddhabrots are not "fully developed". But at theses iterations i don't know if we can compute a complete Buddhabrot (and I don't know if it'll be interesting).
@kram1032: "So you ONLY take those orbits close to the border of the set?" Yes, very close.
"Also, it LOOKS like you colour each and every orbit you find in a different random colour?"
I have "layers" of orbits whose iterations are between two integers. And then I pick a color for this layer (it is not random). The lightness of the color is set according to the density of orbits (not with a linear function of course, we wouldn't be able to see anything..).
For example in the last image, the green is from orbits between 560k and 700k.

In the next version of my software I will be able to color every orbits separately .

[youhn]

Just digging up some beautiful images. That symmetrical Buddhabrot, let me find the current topic on that.

Found! http://www.fractalforums.com/new-theories-and-research/buddha%27s-jewel-a-special-subset-of-the-buddhabrot-symmetrical-about-both-axis!/

Is it the same? Quote from the topic started by billtavis:

The orbits drawn are for points which are very close to the set and yet take a small number of iterations to escape. In other words, near the tips of branches, equally distributed around the set weighted by the size of the branch (the area around the needle has the most orbits drawn). In this case, all the points I drew were closer than 1e-20 to the set and took less than 200 iterations to reach a very large radius. Most points took less than 100 iterations to escape. There's probably a more precise (i.e. mathematical) way to describe the set of points but I don't know what this would be. Maybe someone here on the forums has an idea?

And ...

In the next version of my software I will be able to color every orbits separately grin .

Wouldn't that give a very meshy greyish results ... ?  

[tit_toinou]

Just digging up some beautiful images. That symmetrical Buddhabrot, let me find the current topic on that.

Found! http://www.fractalforums.com/new-theories-and-research/buddha%27s-jewel-a-special-subset-of-the-buddhabrot-symmetrical-about-both-axis!/

Is it the same? Quote from the topic started by billtavis:

Hi ! No It's not the same.

I'm just looking for high iteration orbits because they stay close to points inside mandelbrot for a long time *yet* they escape at some point.
I also found images on the internet showing that high order points inside the mandelbrot might be interesting too !

It indicates that the Anti-Buddhabrot is not ugly, we just misrepresented it by taking a lot of small iterations points that hide the true high order interesting points and also by plotting too many times the same points that belong to the same cycle.

It means that to me, the most beautiful orbits are thoses very close to the border of the mandelbrot set outside (escaping) OR inside (high order).
Some research should be done regarding this !

( By high order I mean very long cycle, z_{n+p}=z_{n}, p is very high. )

Another visualisation of the anti buddhabrot would be to plot only cycles 1 times (and not x times because we run the iterations p^x more times without cycle-checking) OR only plot the orbits before entering in a cycle (that would of course require cycle detection).


Buddha's jewel is about to find points close to the set yet escaping very fast. I think (by intuition) that the points concerned are thoses where the derivative of the distance to the set is very high = high values of the Douady-Hubbard Potential = where the thunderstorm would make his way if the inside of the mandelbrot set was electrically charged = near the " needles ".

You can look at the images at the end of the first pages posted here : http://www.fractalforums.com/images-showcase-%28rate-my-fractal%29/classic-mandelbrot-with-distance-and-gradient-for-coloring, you'll see white needles !

Wouldn't that give a very meshy greyish results ... ?  

Hum no because I would have a lot of points, I would take points that go on for 1 millions iterations before escaping, giving weird shapes (but individually colored) like in the images posted here.

[3dickulus]

That last image (b/w), to me, is the most interesting with intricate delicacy that leads the eye deeper and deeper. 

[tit_toinou]

A friend of mine is also exploring Buddhabrot the way described in this post.
Here's his first image, more is coming

[kram1032]

what kinds of values are you using for these?
If I understood correctly, what you are doing is that you only look at a fairly narrow range of iteration counts, only even beginning to plot them at high values.
So what is the minimum value of iterations you consider in these images? I'm currently trying a classical BBrot image with:

• 1000 pre-iterations
• 10000 steps before the blue channel bails out
• 100000 steps before the green channel bails out
• 1000000 steps before the red channel bails out

And I further restrict the initial selected points such that both the initial point and the first iteration are inside the circle of radius 4 but neither in the main period-1-cardioid nor in the circle of period 2 to ensure that the selected points are at least reasonably close to the border of the set.
Is there anything else you are doing besides that? What are your choices for the various values?

[kram1032]

Here is the not-nearly-as-dramatic-as-yours result of said settings. (Click on image for larger view)

[Caleidoscope]

Wow, it is beautiful, awesome colors and lovely details.