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Here is an picture from the border of the spine with 17 Iterations.

The same area with 18 iterations.

... and with 19 iterations.

It looks like the inner part of this area ist filled with the outer part of the mandelbulb stuff.

20 Iterations:

The stange thing ist, that some areas seems to be not changing, only the cream stuff...

(21 iterations).

This strange cream does not appear on the 2d mandelbrot set ...

(22 Iterations).

... but on higher iterations, you get a sort of spiral ... like the 2d mandelbrot spiral.

(27 Iterations)

nice sequence, feel free to post deeper depths!

great sequence, love it how more and more details evolve !

 :thumbsup1:

... and with 50 iterations, the inner part of the mandelbulb becomes more chaotic (at least in my renderer Gestaltlupe).

Nice sequence :)

I guess, that "chaotic behaviour" is the next generation of dentrites?

The set's area (volume, respectively) grows smaller and smaller to the limiting shape, rahter than bigger and bigger after all...
So, it should be like an additional fold of a new series of shapes :)

First test for a rendering with 9000 Iterations. In the next week (if my animation sequence is ready) i will start the final rendering. 

Nice shots! It would seem that higher iterations add more 'whipped cream' in general (at least covering the detail). I did something like this a while ago too - may as well put it all together. The numbers represent the iteration level. Look at iteration 15-17, there's a whole tier created above the previous layers. You could go inside....

(http://www.skytopia.com/project/fractal/new/other/spikei12b.jpg)
(http://www.skytopia.com/project/fractal/new/other/spikei16b.jpg)
(http://www.skytopia.com/project/fractal/new/other/spikei20b.jpg)
(http://www.skytopia.com/project/fractal/new/other/spikei24b.jpg)
(http://www.skytopia.com/project/fractal/new/other/spikei28b.jpg)
(http://www.skytopia.com/project/fractal/new/other/spikei32b.jpg)
(http://www.skytopia.com/project/fractal/new/other/spikei34b.jpg)

Here's a higher res of one of them:
(http://www.skytopia.com/project/fractal/new/other/green-spikeydonut.jpg)

twinbee: just wondering, wether an overlay of all of them together would allow both hidden and later comming details... :)

Maybe, a glow-ish buddhabrot variant (or in general a transparent one) could help with the look. (You would have to compromise between clearness and transparency, maybe by doing a distance estimation based transparency level from the inside: The closer to the outer edge, the either more or less transparent, depending on wether you look from the inside or the outside :) )

Wow, love your iteration sequence twinbee!

I would love to have a look 'inside' to see how the obscured areas develop with higher iterations.

Although, I would love to develop my own ray tracer, I don't have the time right now - what's the simplest/quickest way of setting up a renderer for the mandelbulb? For linux...

cheers, and keep the incredible images coming!

Very great pictures. This 'whipped cream' is for me at this time the most interesting part in the mandelbulb theory.

I want to know, if this iteration steps convergence (it must, because ther is only limited space in the mandelbulb). The problem is, that higher iterations needs (especially for the inner part) very much CPU-time.   

I think its time for some theoretical research.

On the top of the tower, the whipped cream seems to converge.

http://www.youtube.com/watch?v=DOPnOh-OruA

This reminds me somewhat of how, at the "entrance" to a Julia set (in th regular 2D MSet), the iterations can start to remove very little each time, but eventually they "squeeze" through the "entrance" and sort of "blossom" out again into smoother curves. But they converge anyway. The way that this iterations of the MSet seem to stop at sharp edges and then inflate again into whipped cream seems like a very similar mechanism... would this imply that the 3D tier being built would be some kind of corresponding part to a Julia Set?

Thinking about it and looking at the iteration-sequences:

It looks like (if you look at the inside) whipped cream is simply an area of growth, happening after the fractal details in the previous level of whipped cream got too tiny.
Maybe, this happens after fractal detail reaches a special generally calculable, uhm..., fractalillity, which could possibly help to find "perfect" iteration levels, where most details in the current layers are present.

Maybe, "tools" could be found to "cut" parts out of the set like single nerves from bodies.

In case of the 2D-set, if you imagine a person to have a surgery, who comes from the flatlands, it would be a piece of cake to do so: You wouldn't even have to cut him open as you already see what's on the inside and could directly alter the inside that way.
Surgeries would be as simple as ever possible.

But try to find a single blood vessel in a 3D-human being. - Tools to cut open without harming too much are needed. So, let's develop nice "knife"-tools to lay free structures of the boundary, hidden under other levels of detail...

Actually, what's whipped cream in the Mbulb could be comparable with connective tissue in our bodies - those thin skin-like structures just hold other parts of us together. In between, we have hour organs of different fractality :)

This is going a little off-topic, but what if we could see in 4D?

Improvements for artificial eyes are improving rapidly, and I don't know quite how good they are, but I think I read that they've shown the brain can adapt to correctly interpret the signals being sent by a camera.

Couldn't it theoretically be possible to transmit 4D visual information to the brain?

Couldn't the brain adapt, allowing us to see all four dimensions correctly?

Wouldn't this crazy pwnge?

It would take a very long time, I guess, for the brain to figure it out, but once it did... that's be awesome. Imagine being able to see the entire Julibrot set's surface - no slicing. Imagine being able to see a full quaternion Julia's surface. We could also use that to see the entire Mandelbulb laid flat. So I say: wait 40 years for insanely awesome android eyeballs, then it should be easier.  :D

I'd rather directly go for 5D as many of those sets need 5 Dimensions to be laid flat. :D
You know: It's one way to get the brain to refactor 3D-views as for that it already has a certain fixed (more or less flat) area, but even if you could send 4D-images to the brain, finding a way to adapt the optical field to allow such a projection, how would you get it to learn 4D-movements? As long as it doesn't, all the interpretations will look 3D, as the 4th Dimension doesn't change.

I don't say it's impossible, but you really'd have to play with more than just the visual system. Also the movement-system needs to be affected...

"Easiest" would be to use "3D" glasses which however project 4D-stuff to your eyes and one of those headbands, measuring brainwaves (EEG however not too accurate, afaik) together with muscular "waves" (MEG) and uses that data to control a computer.....

Basically, to "move", through that 4D-environment, you need to "think to move". The computer interprets the data from the brain, the brain interprets the visual data obtained and by that, you could learn :)

However, I really don't know how you would do those 4D-pictures... Just by rotating around it like with a 3D-object, you don't get a full 4D-vision.

You'd very likely need a mix of 3D and 4D images, learn "both" (or are there more) kinds of rotation in 4D and apply those to 3D....


\OT end :)

Yeah, you could use a EEG/MEG to move around, or just a simple keyboard. It's true that by moving around you wouldn't see the entire 4D object - but the same is true for 3D. We can't see the entire Mandelbulb, even with rotation. I think that if we can transmit two sets of 2D information to build up a 3D perspective, we should be able to deliver two sets of 3D information to build up a 4D perspective, don't you think? It would require a pretty fast bus speed for the brain, and a fast computer to be responsive, but it would open up so much if it worked.... can you imagine a 4D MMORPG? Graphic design would be something totally new.  :w00t:

4D perspective isn't actually hard:

Just the same equations:

first two Dimensions are undistorted and the other two have perspective distortion ;)

I once tried to draw 4D perspective hypercubes by hand :)

Worked not too bad. All you need is two points of infinity rather than 1. :)

You could also add perspective to the first two dimensions which in some pieces of art is a trick to let you uncertain about where's up and where's down and such.

AFAIK, an example are those pieces of art of infinite staircases with monsters going everywhere. You can't be sure if you're looking down or straight forward or even up due to perspective irritations :)

\second attempt on OT end :)

Great anim trafassel!


Translucency sounds ace, but will have to wait until I incorporate a scan of more than just the visible shell into my renderer. Maybe one could use a glow based on the surface normal (in relation to the camera, perhaps so that viewing the surface head on will appear least brightest).

I'd like to animate this smoothly, but it would be nice to be able to render a continuous realm of iterations (instead of the discrete/integer amount that the Mandelbulb uses intrinsically).

Cross sections of the above pics would be intriguing too. Sigh, so much to do, renderer and mandelbulb wise!


I doubt we can truly 'grok' 4D very well. We can't really even see 3D properly (otherwise we'd be able to see through and all stuff simultaneously) - it's more 2 times 2D (i.e. 2 eyes x 2D pixel arrays).


Heh, yeah that's cool - is that how 4D is often visualized? 3D is already cramming loads of hacks onto 2D (light/dark shading, perspective distortion, two eyes, blurring etc.), so maybe one or more of those hacks could be 'borrowed' for 4D in a more intuitive way.


Failed ;)

 :'( xD

Well... I guess, as there is no way to have 3 eyes, which, with a shift also in the 4th axis for the 3rd eye in theory would allow 3x2D -> 4D vision, you actually'll HAVE to borrow the 2 eyes :)

Perspective would be extended to just an other point.

light rules like reflection/refraction and stuff would work the same as they already are restricted to the plane between the ray and the surface-normal of the reflecting/refracting object at the point, the ray hits the object (long explanation for a very short meaning xD).
From the optical point of view, 4D probably isn't that much more complex than 3D. The equations stay more or less the same but things (assuming a monte carlo-style raytracer) will need way longer to clear up.

interesting would be some strange reflection and refraction (prism and lens) setups to direct to-us-invisible 4D-rays directly into the 3D world.
would just be a matter of directing rays to a 0-valued 4th dimension. The simple, trivial hyper"plane" (our space) which has the equation (w,x,y,z as coordinates) z=0 :)

Beautiful and very interesting renderings.

Curious to see how higher iterations gradually “builds up new layers” in your renderings Twinbee.
Makes me want to make thin slices of the last render, with the higest iterations.
And theoretical research would be very nice to follow here trafassel!

Keep on iterating!

(Love the OT in this thread as well ;) )

In the first part of my animation i simulate the  continuous realm of iterations by decreasing the escape distance in the iteration itself.

... on the border of the spine.

Great sequence twinbee. If you look very carefully you will see that surfaces are not being added they are being peeled off. The snow effects represent the layer below that has potential to be shed and the iteration and rules have not defined a stable form. Thus it appears to froth when in fact it is evaporating away in a fog.

I'll see if I can get round to doing that soon - would certainly be interesting!

Hmmm.. are you sure? As far I can see, the surface is only adding material, not peeling away (for example see iterations 7-8 and 16-17).

Wow!  Does the concept of a "fractional" iteration make sense?  For example, I'd love to see twinbee's sequence of renders made into a movie that evolved smoothly...  but you'd have to be able to render versions at iteration = 9, 9.01, 9.02, etc.

It already does that; watch the first 20 seconds carefully. It jumps a bit because it wasn't logmapped.

The principle is similar to my own convergent smoothed iterations algorithm, or to the smooth coloring for the Mandelbrot exterior that you sometimes see.

P.S. hooray for whoever got the site restored after some wacko deleted pretty much the entire database (to judge from all my saved URLs giving 404 errors). Looks like you had backups; when there was no change for over a full day I'd feared the worst.

Please do try to keep the forum software up to date from now on -- unpatched forum software is a very common port of entry for site-blankers and various other vandals and miscreants. PhPBB is the worst, but this SMF 1.1.11 is clearly not invulnerable. (I'd also like to know why it took over 24 hours to get the data restored. Oh, yeah, because it took more than 12 just for anyone to get notified that there was anything in need of restoration, because both webmaster and postmaster bounce, in flagrant violation of standard Internet RFCs. Please, someone, fix that too, preferably before the next instance of vandalism so as to be able to get a heads-up about it immediately if and when there is another incident.)

Twinbee: did you notice (I'm sure you did) that the whipped cream reaches maximum levels at every iteration count that is a multiple of 8 (i.e. your shots of the 16th, 24th, and 32nd iteration)? There is probably a nice mathematical reason for this. My question is: does whipped cream reach its maxima at these values everywhere on the mandelbulb (this is the power 8 one, right?). And for the bulbs of other powers, is it the same story or do the whipped cream maxima appear at iteration counts related to the bulb exponent?

Can't the same technique that's used for colouring the 2D mandelbrot with smooth colours (instead of jumps), be applied?

Well, it works with iterations, but not really with colors. Here's the code I used:
VaryingBailout {
;For use with the "Pixel" formula.
final:
 complex x = 0
 complex c = #pixel
 int iter = 0
 repeat
  x=x^2 + c
  iter=iter+1
 until (iter==@Iterations) || |x|>=@HighBailout
 If |x|>=@HighBailout
  #index=0
 Else
  #index=1/(|x|-4)
 Endif
default:
 int param Iterations
  default=8
 endparam
 float param HighBailout
  caption="Large Bailout"
  default=1E50
 endparam
}
Even with such a low iteration counts and such high bailouts... nada. Maybe #index=1/log(|x|) would work better, but I don't have time right now. Just one last note: remove the -4, so that it reads #index=1/|x|, and apply it to the inside for Mandelbrot formula with a iteration count on the order of 100. Produces very nice inside coloring!  ;D

gussetCrimp: Before I answer, one thing to bear in mind is that the smooth effect you see in those pics isn't the "whipped cream" effect I usually refer to. The 'real' insidious whipped cream that plagues the Mandelbulb never sprouts any more detail after further iterations (at least not along all axis the way we'd like). It looks like stretched taffy. However, the smooth bulbous surfaces you notice in my pics will sprout more detail after further iterations as you can see. The real 3D Mandelbrot (if it exists) will also have these.

Even the standard 2D Mandelbrot has these bulbous surfaces with a low enough iteration. See the top right part of this picture for instance:

(http://www.skytopia.com/stuff/fractal/stalkm.jpg)

But anyway to answer your question, I think the answer is a generic 'no', because it depends on zoom level and location of the camera. Explore the 2D Mandelbrot with a finite number of iterations to explore this effect.

For evidence see my original 'Spine' pic which has some sections very smooth, but other sections very detailed:
http://www.skytopia.com/project/fractal/new/full/q85/Mandelbrot-Crustacean-med.jpg

However, there is a chance that given any particular part, the smooth bulbous effect may return according to the power of the fractal (yes it is exponent 8 for my iteration collection).

Okay, I do realise this is a bit Off Topic, but I've found a pretty good coloring algorithim based on the idea of increasing th Bailout. It creates a perfectly smooth gradient for any formula of the type z->z^n+c, and can easily be adapted to any other fractal Escape Time fractal. It also creates a nice, smooth interior coloring that varies a good deal depending on iteration, if you use the Pixel formula; if you want a different interior coloring, use the Mset formula.

Smooth iterations:

http://www.youtube.com/watch?v=yYz3gQ108Mo

How do you know that the increased "detail" is not due to the amplification of errors due to rounding? That's what it looks like to me.

Cooooooool.  I hope that the title "Part 1/3" means that there is a "Part 2/3" and "Part 3/3" in the works.  It's like eagerly anticipating the next Star Wars sequel.  :)

Great! So smooth iteration building is possible. I'm guessing it uses the same kind of idea that the 'smooth colouring' does for the normal 2D Mandelbrot?

http://www.youtube.com/watch?v=qM9fAaZ0_SA

 :worm:
 :worm:
Spectacular!

Why does it appear that higher iterations are adding to the set?  Doesn't iteration normally remove from the set?

Should be removing when sequence diverges, and adding when converges... no?

they simply are considered as belonging to the set, because they diverge after the maximum iteration reached
 :evil1:

when then increasing the iteration they will diverge and thus wont belong to the set anymore...  :police:


 O0

These were views from inside the set looking out, i.e. the outside was solid and the inside hollow.

Hi Everyone,

It was two weeks ago my brother send me an email with the link of Skytopia where I discovered these incredibles images of the mandelbulb world.
Immediatly I told myself: I want to do the same, to build a beautiful mandelbulb and make an animation where I can travel all around the mandelbulb, go deeper and deeper in this world with a good music on the background.
I found this forum where I saw these same images. And I have a question, I found lot of software to generate mandelbulb but I never can't generate a beautiful mandelbulb like Twinbee on the first page of this post or like here:
 http://www.fractalforums.com/mandelbulb-renderings/improved-lighting/
I spend all my nights on it and I can't find the formulas parameters to obtain this kind of romanesco shape.
I work with ChaosPro, and I play with the lights, change some parameters of the mandelbulb formula, I never find the way to increase the interation and obtain the pictures like I see on this forum. When the number of iterations increase the details became more and more complex (it's normal of course) but too much fretful, not the same way I can saw on the Twinbbe's iterations example where all the results are very soft.

So, my question is simple: With which software you work to obtain this deepness on the pictures (it is like some fog-shadow on the background) and to increase iteration and keep smooth details? I know it will be time consuming to obtain these beautiful pictures but I am not afraid to work on it.
Thank you for your help.

Ah, of course.  Easier to see the deep parts that way.

That's exactly how it works - a kind of "bailout extrapolation", if you will. In fact, it works for any fractal whose value of z grows like a^(Power^iter), where a is dependent upon the pixel. This means for Mandelbrot sets, Julia Sets, and (for example) the perpendicular slice of the power 1 MBulb, this works. Or, since newton iteration has quadratic convergence (usually), using power=-2 also gives smooth coloring. It's  very general formula indeed.

P.S. after I read this thread a couple days ago I implemented this formula with several fractal types in ahm.ucl in UF formula library, thinking it was something new. A few hours after uploading I discover the regular "Smooth (Mandelbrot)" did the exact same thing.  :-\

Hi, grundraisin, and welcome to the forum.  The answer to your question is a little tricky.  Many here are using software that they've written themselves.

Some are using UltraFractal with quite a bit of custom "programming" done in the form of plug-ins (for 3D rendering, or lighting, for example).  I don't believe that there's a one-size-fits-all solution to doing this sort of rendering, yet, mainly because the mandelbulb itself is such a recent discovery.

Your best option might be to choose a specific image or animation from a particular member, and ask him or her what software or method was used in that instance.

The same scene with my new renderer (source code available at github, project "Gestaltlupe").

Compile this project on Visual Studio 2008 (or sharpdevelop), load "settings\animation\varIterations\eins.xml" and press start to render one picture.

All 3 parts in one film:

http://www.youtube.com/watch?v=FavM9Y4d4FY

it's like liquid that crystalizes in stunning shapes :D

For smooth iteration in this video i used the following computations. No much math here, just playing with parameters and comparing results.

//Initialisation:
double gr1=GetDouble("Formula.Static.Cycles"); // Number of iterations
int tempGr=(int)gr1;
double gr1Frac=gr1- tempGr;
gr1=1-gr1Frac;
gr1*=2.4;

double gr =Math.Pow(10,gr1)+1.0;  // Bailout

// ... in iteration:

double r= Math.Sqrt(x*x+y*y+z*z);
if (r > gr) { tw = n; /* abort iteration: no inner point */ break; }

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