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Author Topic: True 3D mandelbrot type fractal  (Read 423203 times)
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David Makin
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« Reply #105 on: June 23, 2009, 02:57:30 AM »

Hi, just posted this in another thread - meant to post it here - so here's the link:

http://www.fractalforums.com/images-showcase-(rate-my-fractal)/%27real%27-3d-mandelbrot-attempt/msg7366/#msg7366
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twinbee
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« Reply #106 on: June 24, 2009, 12:15:46 PM »

With the 3rd dimension thrown in, it's like exploring unchartered space ala Star Trek, all this stuff.

David, I liked the way of demonstrating 3D that those 'sliced' animations used. (Wouldn't mind seeing that one or your latest rendered in Lycium's renderer).
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Buddhi
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« Reply #107 on: July 09, 2009, 08:17:04 PM »

Hello
I made some 3D Mandelbrot fractal using David's 4D formula:

newx = xx*xx - yy*yy - zz*zz - ww*ww;
newy = 2.0*xx*yy + 2.0*ww*zz;
newz = 2.0*xx*zz + 2.0*yy*ww;
neww = 2.0*xx*ww + 2.0*yy*zz;
xx = newx +a;
yy = newy +b;
zz = newz +c;
ww = neww;

I rendered this fractal into 3D array: 1500 x 1500 x 1500 pixels and later generate 3D view with some very simple but effective shading algorithm. It takes very huge amount of memory. Only 3,2GB RAM :-) but rendering is very fast. It takes only 1 hour to render this. I write my own program in C++ to render this

First attached render uses iterations between 15 and 255 to make some "fractal fog".

http://www.fractalforums.com/gallery/?sa=view;id=727

Second uses iterations higher than 240 and there are visible very sharp shapes.

http://www.fractalforums.com/gallery/?sa=view;id=728

Thanks Twinbee and David for inspiration. You have done very good piece of work.

For more fractals I invite to my gallery: http://picasaweb.google.com/buddhi1980/Fraktale
« Last Edit: July 11, 2009, 08:33:22 PM by Buddhi » Logged

cKleinhuis
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« Reply #108 on: July 09, 2009, 08:26:13 PM »

wow, the second one is cool, i believe a pertubated mandelbrot .... very nice!
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David Makin
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« Reply #109 on: July 09, 2009, 08:27:15 PM »

Congratulations on the nice renders !

I'm still hoping that Lycium will have a go at this one using his fractal rendering engine wink
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Buddhi
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« Reply #110 on: July 09, 2009, 08:42:27 PM »

When I made my first attempts to generate 3D fractals I use Tweebee's algorithm with calculating (doubling) angle of iteration vector. I think it looks very nice:

http://www.fractalforums.com/gallery/?sa=view;id=725
http://picasaweb.google.com/buddhi1980/Fraktale#5298247532536219042

Next I tried to use 3D rotation matrix to double angle of vector:

calculating angles:
alfa = atan2(y,x);
beta = atan2(z, sqrt(x*x + y*y));
             

create rotation matrix:
                          |cos(alfa)   -sin(alfa)   0      |
Rot z (alfa) =     |sin(alfa)    cos(alfa)   0      |
                           |    0             0              0       |

                           |cos(beta)    0         sin(beta)   |
Rot y (alfa) =     |    0             1             0           |
                           |-sin(beta)   0     cos(beta)      |

coplette double rotation:

Rot = (Rot z * Rot y) * (Rot z * Rot y);

and we have this formula:
r = x*x + y*y + z*z;
cz = cos(alfa);
cy = cos(beta);
sz = sin(alfa);
sy = sin(beta);
newx = r*(cy*cy*cz*cz - cy*sz*sz - sy*sy*cz) + a;
newy = r*((1+cy)*sz*cz) + b;
newz = r*(sz*sz*sy - cz*cz*sy*cy - sy*cy*cz) + c;

And look at results (it looks so strange):

http://www.fractalforums.com/gallery/?sa=view;id=726
http://picasaweb.google.com/buddhi1980/Fraktale#5356525301118658994
« Last Edit: July 11, 2009, 08:35:55 PM by Buddhi » Logged

David Makin
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« Reply #111 on: July 09, 2009, 08:54:57 PM »

wow, the second one is cool, i believe a pertubated mandelbrot .... very nice!

It's not actually a perurbated Mandelbrot, it does start from (0,0,0,0) but it's using unconventional 4D maths to calculate z^2+c.
The maths involved is a 4D commutative ring but not a division algebra and hence not a field.
Standard Quaternions are a division algebra but are a non-commutative ring and hence are also not a field.
Standard Hypercomplex (sometimes called BiComplex) is a commutative ring but only a partial division algebra and hence again not a field (partial division algebra because not all non-zero values have an inverse).
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The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
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Buddhi
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« Reply #112 on: July 10, 2009, 09:35:54 AM »

"Front view" of Quaternions Mandelbrot (iterations between 15 and 255)


http://www.fractalforums.com/gallery/?sa=view;id=729
http://picasaweb.google.com/buddhi1980/Fraktale#5356729663908398674

Next I will try to render MiniBrot in 512 iterations
« Last Edit: July 11, 2009, 08:37:03 PM by Buddhi » Logged

bib
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« Reply #113 on: July 10, 2009, 03:32:22 PM »

This last one is very impressive with the red peaks showing the 2D M-set. If you zoom on these, are they completely flat or do they have some thickness with potentielly new fractal shapes ?
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Buddhi
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« Reply #114 on: July 11, 2009, 08:50:01 AM »

3D Minibrot at coordinates:
x: -1.7877, 1.7847
y: -0.0015, 0.0015
z: -0.0015, 0.0015


http://www.fractalforums.com/gallery/?sa=view;id=730
http://picasaweb.google.com/buddhi1980/Fraktale#5357087816457273234

and something near this Minibrot at:
x: -1.78524, -1.78495
y: -0.00014, 0.00014
z: -0.00014, 0.00014


http://www.fractalforums.com/gallery/?sa=view;id=731
http://picasaweb.google.com/buddhi1980/Fraktale#5357087793505057986
« Last Edit: July 11, 2009, 08:39:05 PM by Buddhi » Logged

Buddhi
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« Reply #115 on: July 11, 2009, 08:54:10 AM »

View inside 3D Mandelbrot
coordinates:
x: -1, -0.60
y: -0.3, 0.1
z: -0.3, 0.1


http://www.fractalforums.com/gallery/?sa=view;id=732
http://picasaweb.google.com/buddhi1980/Fraktale#5357087896354851314
« Last Edit: July 11, 2009, 08:40:15 PM by Buddhi » Logged

Buddhi
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« Reply #116 on: July 15, 2009, 07:38:01 PM »

Another view inside 3D Mandelbrot
region coordinates:
x: 0, 0.5
y: -0.25, 0,25
z: -0.25, 0.25
max iteration number: 256
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stigomaster
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« Reply #117 on: July 17, 2009, 09:03:24 PM »

Hello guys, I've been keeping an eye on this thread for a little while, although I didn't actually join FractalForums before today. State of the art-work like this is really exciting to watch =)
Just a little thought I just got: There is a 4-dimensional object, a kind of julibrot, of which the familiar mandelbrot set and julia sets are 2D-slices of, and the quaternion julias are 3D-slices. As we know, the 4th dimension can be thought of as time, and therefore the entire 4D-object might be presented as an animation of a morphing quaternion julia.
In the same way, you can imagine a movie as a 3D object, as a prism where a 2D cross-section is a frame of the movie and the height represents time. But there is only one right direction to play the movie. Taking the cross-sections from the side would look completely different.
In the same way, how do one know if one is going through the "correct" axis on the 4D julibrot?

I kind of think this is relevant to this discussion, although I can not bring you amazing renders. I don't even know if I can render a sphere in POVray without consulting the tutorial tongue stuck out
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David Makin
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Posts: 2286



Makin' Magic Fractals
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« Reply #118 on: July 17, 2009, 09:28:34 PM »

Hello guys, I've been keeping an eye on this thread for a little while, although I didn't actually join FractalForums before today. State of the art-work like this is really exciting to watch =)
Just a little thought I just got: There is a 4-dimensional object, a kind of julibrot, of which the familiar mandelbrot set and julia sets are 2D-slices of, and the quaternion julias are 3D-slices. As we know, the 4th dimension can be thought of as time, and therefore the entire 4D-object might be presented as an animation of a morphing quaternion julia.
In the same way, you can imagine a movie as a 3D object, as a prism where a 2D cross-section is a frame of the movie and the height represents time. But there is only one right direction to play the movie. Taking the cross-sections from the side would look completely different.
In the same way, how do one know if one is going through the "correct" axis on the 4D julibrot?

I kind of think this is relevant to this discussion, although I can not bring you amazing renders. I don't even know if I can render a sphere in POVray without consulting the tutorial tongue stuck out

Hi, Here's a morphing Julibrot - in this case the 4 fractal dimensions - zstartreal,zstartimag,creal,cimag - are implimented in 3D space by using two of the fractal dimensions directly as 2 spatial dimensions and then using the 3rd spatial dimension as a line in the plane of the 2 remaining fractal dimensions and gradually rotating the angle of this line in this plane by 90 degrees from one of the fractal dimensions to the other, then doing the same with a different spatial dimension:

<a href="http://www.youtube.com/v/gr-ul7sZDwc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/gr-ul7sZDwc&rel=1&fs=1&hd=1</a>
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The meaning and purpose of life is to give life purpose and meaning.

http://www.fractalgallery.co.uk/
"Makin' Magic Music" on Jango
stigomaster
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« Reply #119 on: July 17, 2009, 10:01:19 PM »

HAHA! cheesy I commented on that video on YouTube two days ago!
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