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Author Topic: True 3D mandelbrot type fractal  (Read 262818 times)
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David Makin
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« Reply #420 on: November 11, 2009, 02:47:10 AM »

Quote
Just a suggestion - where you discuss the formula you could also mention Jos Leys' algorithm for accurate analytical DE for the formula (and related formulas such as Garth Thornton's variations and Rudy Rucker's formula).

Hi David, That's an idea, at least a mention would be good. I think Paul's done a good job of collecting the different formulas, so I probably won't repeat the math. But in any case, I may first link to some thread posts until I understand some of them better. Out of interest, how much faster is rendering after using Jos Leys' algorithm compared to before?

Cheers for the formulae Paul. I look forward to implementing it, especially the 4th power one. I take it that one can use it in conjunction with Jos Leys' DE algorithm?

Hi all, just did some accurate timing tests (using my core2duo instead of the somewhat unreliable timings on this P4HT) just to see the difference between my Delta DE and Jos Leys' Analytical DE and also to see how much faster the non-trig versions are.

First to anwer Daniel's question - yes you can use the non-trig versions with Jos' analytical DE *but* at the moment I only know how to do that using the trig version of the formula for calculating the derivative (with the "normal" iteration using the non-trig version). There's probably a way to get the derivative without the trig but it's beyond my maths ability smiley

OK now the timings - all rendered within UF at the same (high) detail settings with shadowcasting and all at the same magnification. In all cases the "entire" Mandelbrot was in view (all renders 1024 wide).

z^2+c

Delta DE (trig): 9 mins 12 secs
Analytical DE (trig): 5 mins 25 secs
Delta DE (no trig): 1 min 44 secs
Analytical DE (trig for derivative only): 4 mins 26 secs

z^3+c

Delta DE (trig): 4 mins 53 secs
Analytical DE (trig): 3 mins 07 secs
Delta DE (no trig): 0 min 53 secs
Analytical DE (trig for derivative only): 2 mins 29 secs

z^4+c

Delta DE (trig): 4 mins 04 secs
Analytical DE (trig): 2 mins 30 secs
Delta DE (no trig): 0 min 59 secs
Analytical DE (trig for derivative only): 2 mins 04 secs

Pretty much as you'd expect based on the relative amounts of calculation involved smiley

If anyone's wondering there was essentially zero visible difference between the trig renders and the non-trig renders - just the odd pixel in the "bowl" areas of the z^3 and z^4 analytical DE renders - probably down to greater inaccuracies in the trig versions as compared to the non-trig versions (the way I calculate the normals is especially sensitive to small variations).
« Last Edit: November 11, 2009, 02:52:08 AM by David Makin » Logged

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bib
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« Reply #421 on: November 11, 2009, 02:12:27 PM »

Hi
Playing with old 3D formulas is still good fun. Some videos are still calculating with intersesting 3D fractal details from pseudo-seahorse valleys...
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David Makin
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« Reply #422 on: November 11, 2009, 03:25:39 PM »

Some Julia Set timings - all rendered within UF at the same (high) detail settings with shadowcasting and all at the same magnification. In all cases the "entire" Julia was in view (all renders 1024 wide) and the seed was (-0.05, 0.8, 0.1).
Basically this just backs up the results for the Mandelbrots.

z^2+c

Delta DE (trig): 9 mins 50 secs
Analytical DE (trig): 4 mins 50 secs
Delta DE (no trig): 1 min 51 secs
Analytical DE (trig for derivative only): 3 mins 59 secs

z^3+c

Delta DE (trig): 4 mins 17 secs
Analytical DE (trig): 3 mins 0 secs
Delta DE (no trig): 0 min 43 secs
Analytical DE (trig for derivative only): 2 mins 23 secs

z^4+c

Delta DE (trig): 3 mins 28 secs
Analytical DE (trig): 2 mins 31 secs
Delta DE (no trig): 0 min 48 secs
Analytical DE (trig for derivative only): 2 mins 06 secs

In the case of the Julias the trig and non-trig versions were all visually identical.
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« Reply #423 on: November 11, 2009, 05:49:17 PM »

Wow, I did not realize that the nonştrigonometric formulas were so much faster. Here are some higher order formulas that I found with Mathematica. Hopefully, these formulas will provide huge time savings, especially for the popular 8th order version. One nice thing about the odd powers is that they do not require any square roots, so they should be quite fast. These formulas are a bit lengthy, but they checked out correctly on my computer. For optimum efficiency, you will probably need to declare some intermediate variables to avoid redundant calculations.


* White Formulas - Higher Powers.gif (5.1 KB, 895x235 - viewed 1738 times.)
« Last Edit: November 17, 2009, 05:49:37 PM by bugman » Logged
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« Reply #424 on: November 11, 2009, 07:05:47 PM »

Another deep zoom on the power 8 Mandelbrot.
Magnification 4E15, so a million-billion times.
Ultrafractal switched to "extended precision" to calculate this. (meaning it worked with numbers of 20 decimals)
Any further and UF starts "arbitray precision" which takes forever to calculate, so I think this is about the limit with UF (unless Dave has other ideas. wink )
Anyway, I guess we are all convinced now that this thing is fractal, and quite uniform, so at whatever magnification we see more or less the same thing.


* 4D_test_93L.jpg (157.82 KB, 640x640 - viewed 2320 times.)
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Daniel_P
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« Reply #425 on: November 11, 2009, 08:35:24 PM »

Hi all,

Awesome work with the Mandelbulb!

My first post here.
I've been wondering for a little while about another possible way of generating a 3D Mandelbrot and would appreciate your thoughts.

A comment in this Reddit triggered this recent line of thought: http://www.reddit.com/r/math/comments/9nnc0/hypercomplex_fractals_pics/c0djtbw
Quote
Given two complex numbers z and w, repeatedly apply the transformation z→z^2+w (while w stays fixed)
you stop whenever the modulus of z becomes greater than 2. Then take the set of pairs (z,w) for which this process never terminates (of course, in reality you put an upper limit, say, 1000 iterations). As a pair of complex numbers has 4 real components, the set is 4 dimensional. This is the full Mandelbrot set.

Since seeing some of Thomas Banchoff's animations years ago I've been fascinated by the 3-sphere and its stereographic projection into flat 3-space (Something I explored in my 4D rotation animations here: http://spacesymmetrystructure.wordpress.com/2008/12/11/4-dimensional-rotations/ )
and what I was wondering is: What would it look like to take the intersection of the set in 4-space described above with the unit 3-Sphere and stereographically project down to 3D ?

Actually it would make more sense to work backwards - to check if a point in 3D space is part of the fractal, first reverse stereographic project like this:

   X  = 2 * x / (1 + x ^ 2 + y ^ 2 + z ^ 2)
   Y  = 2 * y / (1 + x ^ 2 + y ^ 2 + z ^ 2)
   Z  = 2 * z / (1 + x ^ 2 + y ^ 2 + z ^ 2)
   W = (-1 + x^2 + y^2 + z^2) / (1 + x^2 + y^2 + z^2)

Then do the iteration to see if it is in the set.

My questions for you are:
  • Has this been done before ?
  • Do you think the results would be interesting ?
  • How would you recommend getting started?
    I have a little programming experience but have never done any raytracing/raymarching before. From reading some of Inigo Quilez's writing I suspect that the Hubbard Douady potential might be useful here
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David Makin
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« Reply #426 on: November 11, 2009, 11:32:25 PM »

Hi all, my WIP formula is not yet ready to add to the UF formula database but I've tidied it up a lot - not everything works yet (such as re-ordering the axes) but I think most will find it useable (unzip, copy all the text and paste into an open UF fractal window - needs UF5):

http://www.fractalgallery.co.uk/MMFwip3D.zip

The UPR has quick example z^2+c quaternion renders - one solid on Analytical DE threshold and one on point trapping - note that pickover stalks do not produce good results on quaternions, try the other fractal types for those.

Edit: The above file still exists but is obsolete:

I just updated the WIP3D formula, new link:

http://www.fractalgallery.co.uk/MMFWip3D.zip

Again unzip, copy all the text and paste into an open UF fractal window.
This time the UPR is an example render of Pickover stalks (orbit trap) - note that "auto distances" is disabled and the solid threshold is much larger than used for solid on iteration/distance estimate.
Several more options in the formula plus some bugfixes in the analytical distance estimate methods.
This is actually named as a different formula so as not to cause compatibility issues with any renders you've already done smiley
« Last Edit: November 14, 2009, 06:30:45 AM by David Makin » Logged

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David Makin
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« Reply #427 on: November 12, 2009, 04:16:30 AM »

Apologies if you guys already have this, but I just realised a more compact form of the White/Nylander formula - though not necessarily fastest in all cases (e.g. even powers<10), though it is faster than the explicit trig version in UF:

            ztemp = ((r=cabs(zri)) + flip(zj))^@mpwr
            zri = real(ztemp)*(zri/r)^@mpwr + cri
            zj = -imag(ztemp) + cj

ztemp, zri and cri all complex
r, zj and cj all real.
@mpwr real, though using complex is possible with this form.
In case anyone isn't familiar flip(zj) simply produces i*zj i.e. makes zj imaginary.
« Last Edit: November 12, 2009, 04:18:16 AM by David Makin » Logged

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David Makin
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« Reply #428 on: November 12, 2009, 05:08:08 AM »

Apologies if you guys already have this, but I just realised a more compact form of the White/Nylander formula - though not necessarily fastest in all cases (e.g. even powers<10), though it is faster than the explicit trig version in UF:

            ztemp = ((r=cabs(zri)) + flip(zj))^@mpwr
            zri = real(ztemp)*(zri/r)^@mpwr + cri
            zj = -imag(ztemp) + cj

ztemp, zri and cri all complex
r, zj and cj all real.
@mpwr real, though using complex is possible with this form.
In case anyone isn't familiar flip(zj) simply produces i*zj i.e. makes zj imaginary.

It'd be interesting if someone can work out how to apply a full 3D "power" smiley
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iq
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« Reply #429 on: November 12, 2009, 08:00:31 AM »

Hi guys, I just registered to the forum because somebody pointed me to this thread this morning. I got interested in the trigonometric version of the Mandelbrot set posted here, so I wrote a small raymarcher (in C) and tried the idea. Results are interesting. First I tried it in the CPU:



Render time was about 15 minutes (512 ambient occlusion rays, one shadow ray, two light sources, 2x2 antialias at 1280x720). The raymarcher is very rudimentary, the step size is constant, no distance estimation computations to speed the marching. Not yet. The surface normal (if such a thing exists for fractals) is computed by the central differences method, it's not analytic.

Since a couple of years ago I did a demo with a quaternion 4d julia set in realtime with ambient occlusion in the GPU (Kindernoiser: http://iquilezles.org/prods/#kindernoiser), I decided to port this little raymarcher of this morning to the GPU this afternoon. Results are promising, I get almost realtime rendering (5 frames per second) without the distance estimation optimization and some non-raytrace-based ambient occlusion.

My motivation to try this has been one of those green-broccoli zooms posted by Twenbie, it so much looks like real vegetation or a forest, it's amazing. I hope I can work a bit next days again and do something like that, I will continue to do so.

Also, I think twenbie or somebody with a good raytracer handy should try some subsurface-scattering (translucent surfaces), that should rock quite a lot!!!

About the topic, it's really a pity the thing doesn't work that nicely for exponent=2, otherwise I would say this would have been the definite 3D M-Set. In any case, the choice of multiplying both angles by the exponent in the trigonometric multiplication is rather arbitrary I think. It's a pity. I somehow feel the real thing not trigonometric, but polynomial, although I'm not the one to tell. My best attempt so far was this http://iquilezles.org/www/articles/experimental3djulia/experimental3djulia.htm whici is completely ridiculous compared to the beautiful images produced by this trigonometric variation discussed in the thread.

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bib
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« Reply #430 on: November 12, 2009, 10:42:29 AM »

Hi all, my WIP formula is not yet ready to add to the UF formula database but I've tidied it up a lot - not everything works yet (such as re-ordering the axes) but I think most will find it useable (unzip, copy all the text and paste into an open UF fractal window - needs UF5):

http://www.fractalgallery.co.uk/MMFwip3D.zip

The UPR has quick example z^2+c quaternion renders - one solid on Analytical DE threshold and one on point trapping - note that pickover stalks do not produce good results on quaternions, try the other fractal types for those.

Thanks so much David. Many hours of fun exploration to come smileysmiley
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bib
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« Reply #431 on: November 12, 2009, 10:48:31 AM »

Very impressive image for a first post here!
My best attempt so far was this http://iquilezles.org/www/articles/experimental3djulia/experimental3djulia.htm whici is completely ridiculous compared to the beautiful images produced by this trigonometric variation discussed in the thread.

Your 3D julia images remind me a video I did recently :
<a href="http://www.youtube.com/v/uotgvkZO2Zk&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/uotgvkZO2Zk&rel=1&fs=1&hd=1</a>

and this one as well:
<a href="http://www.youtube.com/v/tCGqXDnzje8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/tCGqXDnzje8&rel=1&fs=1&hd=1</a>
« Last Edit: November 14, 2009, 09:03:41 PM by bib, Reason: added the 2nd video » Logged

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« Reply #432 on: November 12, 2009, 12:13:02 PM »

hi (again!) i˝igo, good to see you see you here smiley the render you posted is great!

i was also hoping to try rendering this thing on everyone's favourite little supercomputer, the gpu  afro
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« Reply #433 on: November 12, 2009, 02:28:15 PM »

Hello,   smiley

This very simple iteration can produce some interesting results (in a very unoptimized C code):

bool JuliaSimple(double x, double y, double z){
double radius,ny,nx,nz;
unsigned char Counter;
for(Counter=0;Counter<50;Counter++){

 nx = x*x-z*z-0.27;
 nz = 2*x*z;

ny=y;
x=nz;
y=-nx;
z=-ny;
radius=x*x+y*y+z*z;
if (radius>1000) break;
}
if(radius>1000)
return false; //its outside of the set
 else
   return true; //its on the set
}

These are two other C values:
/*nx = x*x-z*z-0.285;
nz = 2*x*z-0.1;*/

/*nx = x*x-z*z-0.27;
 nz = 3*x*z;*/

here are two renders of this system:
http://www.rfractals.net/gallery/main.php/v/JuliaSets/3DJulia1.html
http://www.rfractals.net/gallery/main.php/v/JuliaSets/3DJulia2.html

The Mandelbrot of this series doesn't looks good but I have tested it on many other julia equations (such as the phoenix and the newton fractal) and it looks interesting.


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Buddhi
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« Reply #434 on: November 12, 2009, 05:40:27 PM »

here are two renders of this system:
http://www.rfractals.net/gallery/main.php/v/JuliaSets/3DJulia1.html
http://www.rfractals.net/gallery/main.php/v/JuliaSets/3DJulia2.html
The Mandelbrot of this series doesn't looks good but I have tested it on many other julia equations (such as the phoenix and the newton fractal) and it looks interesting.

Hello Aexion
It is very nice to see new person in FractalForums. I saw your renderings and they looks very interesting. You have very strange iteration formula. It looks like z^2+c formula but with additional rotation by 90 degrees after each iteration. The results are awesome because there are visible lots of beautiful spirals.
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