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Author Topic: True 3D mandelbrot type fractal  (Read 273600 times)
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twinbee
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« on: September 22, 2007, 11:05:02 AM »

Hi all!
I'm relatively new to fractals, but I have searched for hours to find a true 3D mandlebrot type fractal, all in vain. I don't want the raised mountain type of mandlebrot, and I don't want any true (but trivially simple) ones such as the Menger sponge. I instead want a true 3D equivalent of the mandlebrot (or near enough).

The closest I got was the Quasi-fuchsian sphere fractal:
<a href="http://www.youtube.com/v/3lcO9zRCv-4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/3lcO9zRCv-4&rel=1&fs=1&hd=1</a>

However, that is still mostly self-similar. I want the amazing variety that can be found in the mandlebrot. Anything out there?
« Last Edit: October 19, 2009, 10:26:23 AM by Trifox » Logged
twinbee
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« Reply #1 on: September 25, 2007, 08:16:44 AM »

Thanks so much for your reply!
Even if it wasn't like a 3D rendition of the Mandelbrot, I would love to know of a 3D fractal that has mandelbrot-type beauty and complexity along with the amazing variety. As far as I know, no such fractal exists, but perhaps I haven't looked hard enough?
« Last Edit: September 25, 2007, 09:14:05 AM by twinbee » Logged
twinbee
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« Reply #2 on: September 30, 2007, 10:08:52 PM »

I've just spent an exhausting week trying to create a TRUE 3D rendition of the Mandelbrot (or whatever the equivalent would turn out to be). It's been a really cumbersome, if somewhat exciting journey. Although you say we should be happy with the 2D version (which I am), I believe that maths can do anything, and that it's just a matter of finding the right equations to find the jackpot.

To aid me in my quest for 3D perfection, I have tried to use 3 dimensional numbers, and have even creating my own arithmetic around it. At first I was meddling with the basic 2D formula, but trying to mould it into 3D dimensions using simple algebra (expand the equation and simplify). This leads on to questions such as what you would do if faced with i multiplied by j (j being the 3 dimensional imaginary number). Using solutions based around this approach I got some interesting extruded/distorted 3D Mandelbrot objects, but nothing like what I was hoping for (where the Mandel-type detail is surfacing throughout all 3 dimensions).

Failing that, I also tried to visualize what happens in standard complex multiplications, and used rotation techniques, but in all 3 dimenions, instead of the 2D rotation for complex numbers. In the end, I got another "extruded Mandelbrot" which had a sort of bird shape at one cross section of the 3D object. It was quite interesting, but nothing sensational.

Well that's it. Having done all that, I feel fairly gutted, having failed sad

Still it was quite fun too. If anyone has any inkling whether 3D mandelbrots could even theoretically be possible, I would love to hear.
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David Makin
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« Reply #3 on: October 01, 2007, 12:57:36 AM »

You could probably get what you're aiming for by performing fancy transforms on 3D/4D space prior to plugging coords into quaternionic iteration, though you might consider that as "cheating" smiley
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twinbee
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« Reply #4 on: October 01, 2007, 01:09:31 AM »

I've had a good look over the net and couldn't see anything coming even close. Even quaternionic fractals (as cool as they are), have mandelbrot-depth fractal complexity in only 1 dimension (they look very 'smooth' on the other two axes - which resemble the extruded 3D fractals I was able to create with 'triplex' numbers as I explained in my post above). In fact, they look like whipped cream smiley

No, sorry, I'm after mandelbrot style complexity in all 3 dimensions. Something similar to the Quasi-fuchsian sphere fractal here:
<a href="http://www.youtube.com/v/3lcO9zRCv-4&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/3lcO9zRCv-4&rel=1&fs=1&hd=1</a>
(except that's mostly self-similar, so it's good but no cigar).
« Last Edit: October 01, 2007, 01:18:14 AM by twinbee » Logged
David Makin
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« Reply #5 on: October 01, 2007, 01:44:32 AM »

I know standard quaternionics is no use, but to go back to 2D, you could apply a 2D manipulation of the complex plane (i.e. x and y coords) prior to plugging them into the standard z^2+c iteration to (for example) give the Mandelbrot an extra lobe - or remove a lobe. In the same way you could manipulate 3D/4D space prior to plugging the coords into a quaternionic iteration (for "roundy" results) or a hypercomplex iteration (for "squarey" results) to produce 3D lobes.
As you say though I've seen no-one try this yet smiley In fact I've never seen any 3D/4D fractal software that allows 3D/4D transformations to be applied in that way (i.e. like UXFs are used for 2D in Ultrafractal).
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twinbee
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« Reply #6 on: October 01, 2007, 01:49:36 AM »

Wow, sounds good. Seriously, if you see any picture on the net, then please let us know ASAP!
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twinbee
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« Reply #7 on: November 15, 2007, 05:22:26 AM »

So far, this is the closest I've come to creating a 3D-style mandelbrot with bulbs on all axis. It uses techniques described in my "Meet & greet" forum thread, and orthographic projection with lighter areas at the front.



Sigh, so close, yet so far!
« Last Edit: November 15, 2007, 05:41:12 AM by twinbee » Logged
twinbee
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« Reply #8 on: November 17, 2007, 02:52:35 PM »

Hope the exam went okay. Here's the way to create the object. I'm not sure how Cayley tables would fit in this context, but here's the multiplication function:

double pi=3.14159265;
double r    = sqrt(x*x + y*y + z*z );
double yang = atan2(sqrt(x*x + y*y) , z  )  ;
double zang = atan2(y , x);
newx = (r*r) * sin( yang*2 + 0.5*pi ) * cos(zang*2 +pi);
newy = (r*r) * sin( yang*2 + 0.5*pi ) * sin(zang*2 +pi);
newz = (r*r) * cos( yang*2 + 0.5*pi );

Since in the standard Mandelbrot forumla, the only multiplication is the number multiplied by itself (the square), that's why there's only x, y and z in the above function, and not x2, y2, and z2 aswell. Hope that makes sense.

The addition function for these 3D numbers is as expected:
newx = x1 + x2;
newy = y1 + y2;
newz = z1 + z2;

The main Mandelbrot formula stays the same ("a" and "point" are 3D numbers of course - X, Y and Z):
add(    multiply(a,a)   , point).

Oh and the escape condition is slightly extended over the 2D version like so:
while (x*x + y*y + z*z  <  2.0)

Thanks for doing this. It's gonna be quite exciting to see how it all turns out!
« Last Edit: November 17, 2007, 09:13:23 PM by twinbee » Logged
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« Reply #9 on: November 17, 2007, 05:16:41 PM »

This is cool, I'll be watching this thread.
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twinbee
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« Reply #10 on: November 19, 2007, 05:00:04 PM »

Oh wow that first one - looks ace!! Amazing how mistakes can often result in these things. It looks like a face with the head part being scalped heh smiley

Ah, yes, the sqrt/sqr in the r variable can be cancelled. Is the pic above (which also looks really nice by the way) the equivalent of my 2D ortho projection? Doesn't quite look like it, but then I wouldn't know what to expect from a different angle!

Quote
edit: nono, it's doubling the spherical angle and adding an offset (different ones for phi and theta). hmm i wonder if this can be made more efficient, cuz damn it's slow :|

Yep, in fact, the spherical angle is doubled not just in one direction, but in the other too. So it's more of a spherical 'twist' than rotation. I wish I could sort of rotate in the 3rd direction too, but as you know, only 2 rotations are needed to represent any possible sphere angle. Maybe there is a way, but I don't want to start representing angles with multiple possible ways of XYZ rotations...

Haha, I'm dying to see higher res versions now. To be honest, I know these things render so slowly, but I'm amazed they can render at all. Even plotting the simple pixel projection I did took around 2 hours in C++, so I'm quite impressed really, especially as it's raytraced. More CPU speed in computers would still really be nice though hehe.
« Last Edit: November 19, 2007, 11:11:41 PM by twinbee » Logged
twinbee
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« Reply #11 on: November 20, 2007, 09:37:34 AM »

Quote from: lycium
Quote from: twinbee
Is the pic above (which also looks really nice by the way) the equivalent of my 2D ortho projection? Doesn't quite look like it, but then I wouldn't know what to expect from a different angle!
yup.

Oh so it is?? But that previous grey one hardly looks like the next gold one. Not even slightly symmetric for instance. Are you sure?

Quote
i rendered a quick top-down preview (low precision, different materials):
Excellent! This is definitely a 3D version of the ortho one I did cheesy Can I ask you a favour? Can you render it at higher resolution, and also veer the light source to one side, so that the left part of the fractal is brighter than the right, so that we can see a more '3D' like picture. Maybe also use the grey material too, as it better reflects the subtle blue sky light source.

Quote
btw, that's not simple ray tracing  it's simulating all possible light/surface interactions (based on metropolis light transport),

That's all well and good, but I won't be happy until you render with full brute force photon motion simulation to accurately emulate what really happens. Lol, just kidding... cheesy tongue stuck out

Quote
it's simulating all possible light/surface interactions (based on metropolis light transport), spectral skylight (based on a practical analytic model for daylight) in a completely correct way, which requires 100-1000 or more samples per pixel for clean images...

How does that compare to photon mapping or radiosity out of interest?

Quote
.. for me fractals are just fodder for my first love, rendering systems wink
On a tangent, it's be interesting to experience 3D in general (or this world with special goggles) with inverse perspective, so that nearby objects are small, and more distant objects are big (the larger more distant objects would still be 'behind' the nearer stuff). I bet it would look really cool...


Quote
i don't think we'll ever see a 100m times speed improvement, there are physical limits (if you don't believe this, ask yourself: why stop at 100m? why not 1000000000000 quadrillion times faster?) i'd say we have another 1000x or so to go, at most.

Yes that prediction for 2040 is looking slightly optimistic now (that article was written 2 years ago). It's quite a shame the increases have been so small lately sad At least there's parallel processing which will good for raytracing, and radiosity type effects (I hope).

Quote
oh and about physical limits, do you really think that the speed of light will be broken in the year 2600?
Haha tongue stuck out Yeah it probably won't, unless we manage to cheat space-time somehow. Haha, one can dream cheesy cheesy


Quote
it's also definitely the mandelbrot set, here's a thin slice (from y = -0.15 to +0.15) of the object:

Nice!! I'd die to zoom right into that top-left little cave part next to the spindly part to see how much of the detail has survived the move to 3D. If you have time of course. I don't want to eat up all your CPU time! wink

Quote
and doing analogous transformations in the other spherical co-ordinate; because of this there are actually entire families of 3d mandelbrots, depending on your choices of angle coefficient and offset.

I wish!... As good as it looks, this beast is far from the real thing, because most of the infinitely complex details haven't survived the Z axis. Unfortunately, they still look as though they've been 'smeared' over or lost completely. However, there's a chance some of the infinite detail has nearly survived in smaller portions of the set. For example, that cave part near the top looks as though there could be some interesting stuff. Again, I doubt it though.

Quote
moreover, i've been thinking of visualising the whole 4d object. yes, 4d: actually, at each point in space there is a scalar potential, which can be interpreted as density (like a cloud). since i plan to simulate atmospheric physics (in particular rayleigh scattering - which is what's responsible for blue skies and red sunsets etc.) anyway, this will serve as a really nice volumetric dataset. i'll definitely pre-render it to a grid though, because it's SO damn slow! that ought to look pretty cool, and would actually be way faster than the method i'm using here.

Ace. Can't wait to see some more pics of this thing!
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David Makin
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« Reply #12 on: November 21, 2007, 01:04:35 AM »

here's a preview, i'll update it as it gets smoother:

That render doesn't half remind me of this:

<a href="http://www.youtube.com/v/jeVDBvE5Bg0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/jeVDBvE5Bg0&rel=1&fs=1&hd=1</a>
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David Makin
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« Reply #13 on: November 21, 2007, 01:05:48 AM »

edit: nono, it's doubling the spherical angle and adding an offset (different ones for phi and theta). hmm i wonder if this can be made more efficient, cuz damn it's slow :|

Maybe there's a way of working out a distance estimation for it ?
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twinbee
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« Reply #14 on: November 22, 2007, 05:28:36 PM »

Quote
they are totally equivalent: radiance (which is the radiometric quantity corresponding to what we see) is invariant along directions, so it doesn't matter whether you simulate light going from the eye to the lights (path tracing), from the light to the eye (light tracing), or both (bi-directional path tracing). what i'm doing there is a 100% accurate simulation of diffuse light transport, which is really easy actually.

Wow really? I can't really get one up on that one then smiley I'm a bit surprised, because it took 100 Sun SparcStations 1 month to generate some of those pics, but then it was back in 1991...
Does your algorithm even do exotic stuff like account for red/blue shift for rainbow effects?

Quote
photon mapping is a full global illumination algorithm, in that it can simulate all modes of light transport. however it is biased (roughly, you can't be sure it's converging to the right image)

How about if you throw more CPU time at it. Is accuracy increased proportional to the CPU time?

Quote
the core algorithms used in my renderer are pretty much state of the art, and i'll be extending its functionality a lot these holidays

Sounds ace. I have to say, some of these, although grainy, do look very realistic.

Quote
i've not heard of inverse perspective before, can you explain how it works?

Usually, in 3D, lines of perspective converge to a point. Orthographic images on the other hand don't - everything in the back is just as big as everything at the front. Now imagine one step further than even this. lines of perspective would fly away from each other (actually they would converge, but only behind the viewer).

Parallax would look really strange and interesting like this. I was semi-surprised to see the idea is already out there on Wikipedia.

Still love to see a zoom into that top left cave here: http://www.fractographer.com/wip/3dmandel_slice.jpg
Even if you have to forego some of the more realistic lighting effects for quicker speed, I'd love to get a closer eye on the detail.

By the way, what's the building block of these images? Billions of little cubes? Polygons? Pure mathematical curves? I can't imagine the latter, since the difficulty in translating from fractal to curve is daunting (at least for me).

Quote
that same accuracy extends to less simple scenes, and if you want to see what i'm chasing see http://www.maxwellrender.com/

What does that have, that your algorithm lacks?

Quote
That render doesn't half remind me of this:

<a href="http://www.youtube.com/v/jeVDBvE5Bg0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/jeVDBvE5Bg0&rel=1&fs=1&hd=1</a>

Hi David, still love to see that quaternion technique at creating the 3D Mandelbrot set. Any attempts so far?

Meanwhile, here's another 2 failed attempts. As before, brightness represents the Z axis.


And the other...
« Last Edit: November 22, 2007, 07:14:59 PM by twinbee » Logged
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