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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 ;D).

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.

(http://i288.photobucket.com/albums/ll174/BillTavis/BuddhaJewelFULLRES_zpsbrveiaop.jpg)

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

(http://i288.photobucket.com/albums/ll174/BillTavis/pointsDrawn_zpsjtvsglh3.png)

nice  :beer:

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/ (http://math.ucr.edu/home/baez/roots/).

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!

Nice work!  O0
It reminds me a bit of some work I did on Mobius maps: https://sites.google.com/site/tomloweprojects/scale-symmetry/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.

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?

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


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.


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:

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.

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 (http://people.cs.kuleuven.be/~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"

The definition of the Mandelbrot polynomial is given here (http://people.cs.kuleuven.be/~piers.lawrence/coding/mandel.html). 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).

Here's some closeup renders!
First, a shot of the needle:
(http://i288.photobucket.com/albums/ll174/BillTavis/BuddhaJewelNeedle_zps6lgymw1w.jpg)

Now a close look at the origin: (this took forever to converge on a decent image!)
(http://i288.photobucket.com/albums/ll174/BillTavis/BuddhaJewelOrigin_zpsg0rppaok.jpg)

And finally, an area in the lower left of the full render above, which I am calling "the rolls":
(http://i288.photobucket.com/albums/ll174/BillTavis/BuddhaJewelRolls_zpsk52de8xg.jpg)

dude, the last pic is marvellous!

this looks so 3d ish

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

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


:^)

EXCELLENT WORK!!!

 :beer:

I have to second that observation!

:^)

FWIW, this post inspired me to create my vary first BuddhaBrot:

(http://nocache-nocookies.digitalgott.com/gallery/17/11687_02_04_15_12_33_36.jpeg) (http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17279)


:^D

so, this is a classic buddhabrot or using billtavis technique ?!  :hmh:

gosh, those buddhabrots are full of interesting stuff and your coloring is amazing ... looks like this year is standing under the light of the BUDDHA :D


i see you posted it in the gallery, i edited your post and included the link to the entry   :police:
http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17279

so, regarding buddhabrots in general,
i am a bit disappointed that you dudes do not render different formulas (and hybrids) as buddhabrots ;)

especially using the fine adjustments introduced by billtavis a chest has opened up to bring buddhabrots to a whole new life ... who is in for making a user interface for a nice buddhabrot renderer !?  :fiery: :fiery: :fiery: :fiery: :fiery: :fiery:  :hurt: :fiery: :fiery: :fiery: :fiery: :fiery: :fiery:

 :w00t:

stunning images!  :beer:

I've looked at ker2x's buddhaGenerator, to be more specific the one on github by Emilio Del Tessandoro and managed to get some interesting images but none like those presented here.

I wonder if anybody ever tried folding a Buddhabrot into a Buddhabox or a Buddhabulb?

It's based on the same formula as the Madelbox/bulb  (z=z^2 + c), so unless I am missing something  (and I do tend to miss things when making these suggestions!) it should be possible   O0

Very nice images!  Got me looking through my old software to see what Buddhabrot Generators I may have already...

That's what I've been doing for a while here
http://kram1032.deviantart.com/gallery/8114987/Fractals
But I haven't done any new ones in quite some while. I think I need to write a new renderer at some point. The one I used - modified from an already working one I found running in Processing way back when - is really slow.

Really nice pictures in this thread, especially the last couple since the one that, I must third, looks almost 3Dish.

i once did a hybrid buddhabrot generator but cant find any sources, the buddhabrots are existant for any escape time formula, it is just a way to visualise the iteration process, yes, the buddhabox or the buddhaship (lols) they are all existant, the time seems ripe now for them to come out to the surface

This is basically a classic Buddhabrot. I am not using billtavis' technique, yet...  ;^)

I actually iterated all of the points in the plane here and for each iteration I found the corresponding pixel and added some color's.

Also, I when a pixel escapes, I added some more color to the pixel. One more thing, I did not use any orbit traps.

FWIW, here is a zoom on the last rendering of the Buddhabrot I created:

(http://nocache-nocookies.digitalgott.com/gallery/17/11687_03_04_15_9_25_11.jpeg)

It seems I lost a bit of information here. It has be due to the fact that the zoom basically cropped some of the iterations.

Humm...

As I understand it, the classic Buddhabrot plots iterates only of points that escape - points that don't escape are not plotted.  The anti-Buddhabrot plots iterates only of points that don't escape - points that escape aren't plotted.  Looks like you plotted both with different weights, giving some kind of hybrid "dark matter" Buddhbrot?

I wrote a post about the Ultimate anti-Buddhabrot, which plots only the limit-attractors of points that don't escape:  http://mathr.co.uk/blog/2013-12-30_ultimate_anti-buddhabrot.html  I did some other experiments around that time (late 2013) using something very similar to billtavis' technique, no pics online at the moment though (and will take me some time to trawl my hard drive to find them to upload, might instead take a photo on Sunday of the print hanging on my parents' wall...).

My technique was to trace (using Newton's method) external rays at pre-periodic external angles inwards towards the Mandelbrot set, stopping when the iteration count reaches a certain limit - then plotting all the iterates.  Pre-periodic angles land at Misiurewicz points, and for some reason still unknown to me they are much quicker to trace than periodic angles which land at cusps and bond points.  The other difference from billtavis' method was the colouring - I plotted iterates in colours corresponding to the external angle (in HSV colour space, the angle would be the hue).

If you don't split the rays into periodic and pre-periodic subsets, the pre-periodic part totally dominates - "most" angles are pre-periodic, and in fact any binary floating point representation will be pre-periodic (apart from the period 1 ray at angle 0).

Some old attempts at 3D buddhabrot

http://www.youtube.com/watch?v=iPp781NtSFQ

so, who wants to try a buddhabox ?

Well, if were going to do a 3d Buddhabrot, I would find every corresponding voxel during iteration of the center slice (z-axis of zero) of the Mandelbulb, and start adding color. It is basically the same as 2d, except you use an extra dimension when mapping a point Z to its pixel.

Humm...

Actually, one would need to make sure that the z-axis goes above, or lower than zero during iteration. Now that I think about it, the center slice at z-axis 0 might not ever change from zero. Keep in mind that I am not completely skilled in the art of Mandelbulb iterations.

;^o

Yikes! For some reason I missed the point where a classic Buddhabrot only takes note of iterations that escape!

:^o

Anyway, your "anti-Buddhabrot" looks very similar to the one I did.

Thank you so much for the information Claude!

:^)

@people developing a buddhabrot renderer, is there one to make a buddhabox ? (lol i know i am getting on your nerves .. :D )

Buddhabrot are very everything-intensive.
In most "classic rendering" (non-buddha) fractal you can have 1 thread per pixel, which is very GPU-friendly and also memory friendly (and that's why it's gpu friendly, the memory is a massive bottleneck on gpu).

In a buddhabrot you have to do the usual stuff, to test if a point escape or not (feel free to chose to plot a buddha or an anti-buddha after that), which is the easy part.
Now you have to eventually plot millions of point on the whole memory space in a pure chaotic order.
It's a 100% guaranteed cpu cache miss : ssssssssssloooooooooow.
And in theory you have to use an annoyingly global lock that kill the multi-thread performance.
In buddha++ we choosed to removed this mutex because it made no visible difference.

We used the intel profiler (free for opensource project on linux (not for windows).

Then you have to do the rendering, which require to know the max value of a pixel, and reprocess every pixel for a nice rendering.
(And there is no way to do that without a global lock this time)

And we use the metropolis-hasting algorithms to make everything so much faster when you start to zoom.

That's another problem when you're zooming on a buddhabrot compared to a mandelbrot :
- On a mandelbrot, if you zoomed on [0,0.1] and your resolution is 1000x1000 you're going to compute 1 million time in the [0,0.1] space and your job is done.
- In a buddhabrot you have no way to predict where your orbit will go. So even if you zoomed on [0,0.1] you still have to do the computation in the [-2,0.5 ; -1.3,1.3] space billions and billions and billions of time, twice. The more you zoom the less chance you have for an orbit to be visible in your screen. And you can't know it until you did the whole cpu intensive computation. The metropolis-hastings algorithm is helping and get good result even if it's less than perfect (there is a statistical chance that the neighbor of a point visible on your screen will be visible too).

And contrary to the classic mandelbrot, it can use an insane amount of ram.
A classic mandelbrot don't need much ram and adding more ram isn't going to help you in any way, using more cpu is easy and 100% efficient (you can use 1 thread per pixel and each pixel doesn't interact with other pixels).
In buddhabrot, the more ram the better and multithreading is much more difficult because it require some global lock here and there.

That's why we're focusing on z^2+c, it's fast, efficient, well known, and give good result. And there is still discovery to be done on this one before switching to something much more complex and even more everything-intensive.  :angel1:

In my experiments trying higher order polynomials, a couple of things stood out. Instead of spirals and closing patterns, if you go off-center (i.e. if your polynomial has a root not on (0,0), you'd get patterns that look like multiple sheets ending in straight lines. A lot of the patterns do seem like actual sheets being slightly twisted in various ways. I'm sure you can find that if you look at various of my pictures.

I did never attempt to study this in detail though. The above is just roughly my intuition. It basically looks like you are folding the 2D plane onto itself and you get things like mild to strong wrinkles and slightly shifted layering and such.

attached, note that I didn't give it a nice name like "Buddha's Jewel" at the time, I used something like "pre-periodic buddhabrot variation"

IMVHO, a Buddhabrot wrt multi-threading, can benefit from fine-grain locking techniques. Also, if a pixel is a word, why not use atomic fetch-and-add on the pixel value and add some cache padding in the pixel layout to prevent false-sharing?

I just happen to know a thing or two about designing multi-threading synchronization.

State-of-The art:

http://www.1024cores.net/ (http://www.1024cores.net/)

Link is here:

http://www.1024cores.net/home/lock-free-algorithms/links (http://www.1024cores.net/home/lock-free-algorithms/links)

(he has a link to one of my older sites near the bottom of the page "AppCore")


A nice multi-producer/consumer queue I helped develop:

https://groups.google.com/d/topic/lock-free/acjQ3-89abE/discussion (https://groups.google.com/d/topic/lock-free/acjQ3-89abE/discussion)


And a proxy garbage collector:

https://groups.google.com/d/topic/lock-free/X3fuuXknQF0/discussion (https://groups.google.com/d/topic/lock-free/X3fuuXknQF0/discussion)

http://pastebin.com/f71480694 (http://pastebin.com/f71480694)


Not sure if this is relevant at all, but I felt a reason why this just might be of interest to you.

;^)

Thanks everyone!
There it is again! Awesome coloring too! It sounds like you were also plotting the orbits of points just outside the "tips", although you had a more orderly manner of choosing them... and maybe you plotted other points too, like near the centers of spirals and nets?

Yes, it is :)

I hadn't time/energy to code since a long time but i'm switching job from sysadmin/dba/architect/bigdata/coder to just dba, i'll have some time to do weird stuff again  :love:

The link to my old crappy site, can be found here:

http://www.1024cores.net/home/lock-free-algorithms/links (http://www.1024cores.net/home/lock-free-algorithms/links)

(near the bottom of the page "AppCore", under the "source code to study" section)

I subscribed to the lockfree list a few years ago but i'm not reading it :(
But it's all archived so i will, someday :)

FWIW, I whipped up a multi-Julia iterated function system and got a rendering that, IMVHO, has a 3d feel about it.

BTW, when I say multi-Julia, I mean using multiple Julia seed points and dynamically switching between during iteration. The switching is based on a probability factor...

(http://nocache-nocookies.digitalgott.com/gallery/17/11687_15_04_15_2_45_16.jpeg)

Great!  BTW, thank you for taking an interest in to more "exotic" synchronization algorithms out there...

;^D

Fractal-splosion O:

FWIW, here is a result I got from feeding my Buddhabrot implementation with points generated by my hybrid multi-Julia iterated function system. It seems to have a Mandelbrot'ish outline in white, bright pinks and some purples. Humm...

(http://nocache-nocookies.digitalgott.com/gallery/17/11687_17_04_15_12_28_54.jpeg)

FWIW, here is another Buddhabrot fed with points generated by my multi-Julia iterated function system:

(http://nocache-nocookies.digitalgott.com/gallery/17/11687_18_04_15_8_48_41.jpeg)

I have to admit that I am proud of this one!

:^D

Spectacular! ... in a cosmic sort of way  :beer:

Hi there,

Good looking images ;) .
Like I said in this post (http://www.fractalforums.com/images-showcase-%28rate-my-fractal%29/buddhabrot-t10649/msg82839/#msg82839), I think the points you use to plot to the Buddhabrots are the ones that are white in my images (high derivative of the distance <=> high potential) : Mandelbrot with DE (http://www.fractalforums.com/images-showcase-%28rate-my-fractal%29/classic-mandelbrot-with-distance-and-gradient-for-coloring) (at the end).

One thing about the symmetrical thing : the Buddhabrot IS symmetrical since as all complex polynomials have the property P(conjugate(z))=conjugate(P(z)). It just requires a lot of points to notice that (my renders of Buddhabrot don't show this because there is not enough points, but like someone said you can just plot the conjugate orbits and get something symmetrical at the end ;) ).

A zoom on my last Buddhabrot, centered on the main cardioid:

(http://nocache-nocookies.digitalgott.com/gallery/17/11687_19_04_15_10_14_31.jpeg)

I think I might be falling in love with these Buddhabrot's!

;^o

Thanks! And yeah, that seems like a good way to describe the points
Well, like the Mandelbrot set, buddhabrots are symmetrical across the real axis, but not the imaginary axis. It is only a certain subset of orbits that form symmetry across the imaginary axis.

Ok you're not talking about exact mathematical symmetry, right ? The images you have given in the first post show approximate symmetry about the imaginary axis (look at the area you called "the rolls", it is not copied on the other half of the buddha)

that's correct, the images I posted are not perfectly symmetrical across the imaginary axis, although they are very close to being so. The areas which break the symmetry the most are elephant valley and seahorse valley, where higher iteration orbits are found even at the branch tips. It's not clear to me how to isolate the exact symmetry, although it is apparent that the symmetry starts decreasing rapidly for orbits with above 100 iterations.