Buddha's Jewel: a special subset of the Buddhabrot, symmetrical about both axis!

[billtavis]

This hidden gem is present in every standard Buddhabrot rendering and uses the usual z = z^2 +c formula ... but its surprising symmetry about the imaginary axis is usually drowned out by higher iteration points (and so has been missed until now as far as I know ).

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? I will post a grayscale image below which shows the location of points whose orbits were drawn. And the real question of course is: why do these orbits create symmetry both ways? Are these the only orbits with that property? What are other subsets of points with interesting properties?

In the rendering below (in Buddhabrot orientation with -Real pointing up), all points are drawn in the green channel, points that took over 80 iterations to escape are drawn in the blue channel, and points that took over 100 iterations to escape in the red channel. The three colors channels were individually equalized by hand in Krita (using curves) but no other post-processing was done. The view below is a window 4.5 units on each side centered on the origin. If you look closely, you can see the general locations of elephant valley and seahorse valley, the two areas which break the imaginary symmetry the most. 16,000,000 million orbits matching the properties above were drawn in this render.



and here is a cropped view of the distribution of points used:

[quaz0r]

nice 

[laser blaster]

Nice! I see some IFS-looking shapes in there. It reminds me of fractals generated by finding polynomial roots, like this: http://math.ucr.edu/home/baez/roots/.

[cKleinhuis]

ehm, i even see a shading of a real 3d brot there the shades are great, they seem to form a larger structure, or a finer grained structure with at least stuff in the black area but to me it looks 3dish--. great!

[Tglad]

Nice work! 
It reminds me a bit of some work I did on Mobius maps: https://sites.google.com/site/tomloweprojects/scale-symmetry/mobius-maps
like the 3rd from bottom in that link, yours has a bit of a 3D feel to it which is interesting.

[kram1032]

So what you are saying is, you only iterated points right at the border of the BBrot to generate a doubly-mirrorsymmetric BBrot Jewel?

Looks good!
What happens if you try to use points even closer to the border? Do you get some finer details?

[billtavis]

Thanks everyone! I'll have some close-ups to post soon, to have a better look at some of those 3d/IFS looking shapes.

So what you are saying is, you only iterated points right at the border of the BBrot to generate a doubly-mirrorsymmetric BBrot Jewel?

pretty much, but not just any point near the Mandelbrot boundary will do. The point must also escape in a low number of iterations. This means that branch tips will have lots of valid locations around them and valleys will not. You can see this in the distribution map above where the brightness indicates how many points were sampled from each pixel.

Looks good!
What happens if you try to use points even closer to the border? Do you get some finer details?

Thanks! Like the full Buddhabrot, the details are infinite, but also like the Buddhabrot they take a while to appear if you try to zoom in on them. The points I sampled are actually as close to the border as I could get without switching to arbitrary precision. Some points were as close as 1e-26. Some rough pseudocode for how I found the points to iterate:

Code:

angle = uniform random # from 0 to 2pi
radius = 4.0
point = (radius * cos(angle) , radius * sin(angle))
previousPoint = (0,0)
while ( point is outside Mandelbrot and point does not equal previousPoint ) 
     previousPoint = point
     direction = direction towards Mandelbrot set (derivative of the potential)
     distance = distance estimation to closest point on Mandelbrot set (always slightly short)
     point += distance*direction
return previousPoint

Basically, I start with a point on a circle outside the set, then I walk that point towards the set until it can't go any further (my stopping criteria needs improvement - some points get stuck in an oscillation between two locations, for example). This works for finding low iteration locations because of the DLA phenomenon, where branch tips are more likely to be hit, and prevent points from entering valleys.

[knighty]

Cool and pretty picture.
I did once a picture of iterting mandelbrot roots and gess what? looks the same as yours. here is a picture using 23rd mandelbrot polynomial roots (that is more than 4million roots).
See also the animation here by Piers Lawrence.

[billtavis]

Cool and pretty picture.
I did once a picture of iterting mandelbrot roots and gess what? looks the same as yours. here is a picture using 23rd mandelbrot polynomial roots (that is more than 4million roots).
See also the animation here by Piers Lawrence.

Cool! That definitely looks like the same symmetry! So the tips of branches must correspond with polynomial roots in some way? I don't really understand what that means "23rd mandelbrot polynomial roots"

[knighty]

The definition of the Mandelbrot polynomial is given here. the roots are the "centers" of the hyperbolic components. They are always inside the set. Their distribution is very similar to the picture you gave of the points you used (which are obviously ouside the set).

[billtavis]

Here's some closeup renders!
First, a shot of the needle:


Now a close look at the origin: (this took forever to converge on a decent image!)


And finally, an area in the lower left of the full render above, which I am calling "the rolls":

[cKleinhuis]

dude, the last pic is marvellous!

this looks so 3d ish

[youhn]

Hell yeah, specially like that last one. It's a bit IFS-ish. 

[Chris Thomasson]

Cool and pretty picture.
I did once a picture of iterting mandelbrot roots and gess what? looks the same as yours. here is a picture using 23rd mandelbrot polynomial roots (that is more than 4million roots).
See also the animation here by Piers Lawrence.

Wow. FWIW, this look eerily similar to  a rendering I got from an experimental Julia set IFS algorithm I am working on:

http://fractallife247.deviantart.com/art/Crater-Peaks-489725601


:^)

[Chris Thomasson]

Here's some closeup renders!
First, a shot of the needle:
<Quoted Image Removed>

Now a close look at the origin: (this took forever to converge on a decent image!)
<Quoted Image Removed>

And finally, an area in the lower left of the full render above, which I am calling "the rolls":
<Quoted Image Removed>

EXCELLENT WORK!!!