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Author Topic: True 3D mandelbrot type fractal  (Read 372248 times)
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David Makin
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« Reply #120 on: July 17, 2009, 10:13:37 PM »

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

So you did ! I've a terrible memory for (new) names smiley
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David Makin
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« Reply #121 on: July 18, 2009, 01:37:09 AM »

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

I just re-read your post above and I think you may be slightly confusing Julibrots and Quaternions - Julibrots are 4D because they're made up of 2 complex numbers, zstart and the constant whereas quaternions are 4D becuase the zstart in a quaternion Mandelbrot is 4D or because the constant in a quaternion Julia is 4D - one could consider viewing Quaternionic Julibrots but these actually have 8 dimensions - 4 from zstart and 4 from the constant.
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David Makin
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« Reply #122 on: July 18, 2009, 02:26:53 AM »

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".
<Quoted Image Removed>
http://www.fractalforums.com/gallery/?sa=view;id=727

Second uses iterations higher than 240 and there are visible very sharp shapes.
<Quoted Image Removed>
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

Hi Buddhi - I'd be interested to see what rendering this one:

http://www.fractalforums.com/images-showcase-(rate-my-fractal)/%27real%27-3d-mandelbrot-attempt/msg7366/#msg7366

is like when rendered to a fixed grid smiley
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« Reply #123 on: July 18, 2009, 09:26:46 AM »

Do you mean that this
<a href="http://www.youtube.com/v/Sv0x3vOp_Lc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Sv0x3vOp_Lc&rel=1&fs=1&hd=1</a>
is not 3d slices of the julibrot, but a completely different object? I didn't know, thanks.

Just another thing:
At an early point in this thread, someone suggested using the exponent in z -> z^2+c as the 3rd dimension. I realise it's not what we're looking for, because the resulting object has no border, but I'm curious how the 3D render looks like. I've only seen animations like this:
<a href="http://www.youtube.com/v/3Zg3_J_jHCY&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/3Zg3_J_jHCY&rel=1&fs=1&hd=1</a>
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David Makin
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« Reply #124 on: July 18, 2009, 03:40:28 PM »

Do you mean that this
<a href="http://www.youtube.com/v/Sv0x3vOp_Lc&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Sv0x3vOp_Lc&rel=1&fs=1&hd=1</a>
is not 3d slices of the julibrot, but a completely different object? I didn't know, thanks.

From the comment given on that one (and it's appearance) I think it's just a standard Quaternionic Julia morph where the 4D value of "c" in z^2+c where z and c are quaternionic is being modified over time.

For quaternions q^2 expressed as reals is as below:

newx = x^2 - y^2 - z^2 - w^2
newy = 2*x*y
newz = 2*x*z
neww = 2*x*w

You can see from the above that if any 2 of the y, z or w components are always zero then for q^2 the quaternion collapses to standard complex - in fact this is true for any quaternionic maths.
Quaternions can be considered as having one 1D real component "r" and one 3D imaginary component (i,j,k).

More quaternion maths here:

http://en.wikipedia.org/wiki/Quaternion_group
http://mathworld.wolfram.com/Quaternion.html

Also this (but I didn't actually check everything was correct):

http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/arithmetic/index.htm

If you want to delve deeper into 4D number systems then you should also look into the various meanings of "Hypercomplex":

Generally:

http://en.wikipedia.org/wiki/Hypercomplex_number
http://mathworld.wolfram.com/HypercomplexNumber.html

More on the specific hypercomplex numbers mentioned on the Mathwoirld page here:

http://home.comcast.net/~cmdaven/hyprcplx.htm
« Last Edit: July 18, 2009, 03:52:58 PM by David Makin » Logged

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« Reply #125 on: July 23, 2009, 01:37:49 PM »

Have anyone tried using z.r or z.i as the 3rd dimension? That would be the "true" 3d Mandelbrot in the sense that it is a 3d slice of the Julibrot just as the M-set is a 2d slice of it. There is, of course, the problem that the resulting object would be without a finite boundary, but I think it would be interesting anyway. You can i.e. clip it from -1.5 to 1.5. I have made some animations to try and imagine what the 3d version would look like, like this:
<a href="http://www.youtube.com/v/54wDwErEw-Q&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/54wDwErEw-Q&rel=1&fs=1&hd=1</a>
Some interesting things to notice is that the "valleys" stay in the same place as the set morphs, and that some parts are ripped loose from the set only to be reunited with it at a later point. Just pay attention to the "tentacles" at the top.
Will anyone do this? Please?  cheesy
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David Makin
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« Reply #126 on: July 23, 2009, 04:02:08 PM »

Will anyone do this? Please?  cheesy

Ermmm - that's exactly what my Julibrot animation on YouTube that I posted earlier does (the one you commented on), pause the video and look carefully at the first frame and you'll see that frame is essentially made up of the 2D slices from your animation.
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« Reply #127 on: July 23, 2009, 04:07:43 PM »

Yeah, I see it now. Well thanks then! I hope I haven't disrupted this thread with my n00bery.
I also found a render of the same shape (by you) on DA, and I copied the parameters into UF, but when I changed the parameters nothing happened.
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David Makin
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« Reply #128 on: July 23, 2009, 07:33:35 PM »

Yeah, I see it now. Well thanks then! I hope I haven't disrupted this thread with my n00bery.
I also found a render of the same shape (by you) on DA, and I copied the parameters into UF, but when I changed the parameters nothing happened.

If you like mail me privately about it:

makinmagic@tiscali.co.uk
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David Makin
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« Reply #129 on: August 02, 2009, 04:24:12 PM »

Here's another render of the "True 3D" Mandy, rendered using my improved WIP formula for Julibrots/Quats/Hypercomplex etc. with solid based on directional distance estimation (using deltas rather than the derivative) (time 45mins @3840*2880 on a 3GHz P4HT in single-threaded mode) - max. iterations allowed was 60.



If interested you can check out how much better it is than using my older formula which was basically just solid based on iteration:

http://makinmagic.deviantart.com/art/quot-True-3D-quot-Mandelbrot-125190781
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« Reply #130 on: August 02, 2009, 08:18:39 PM »

Hi David
You made very good and accurate render. I'm very impressed by very high resolution of this render. Using your render algorithm you don't have to use large amount of memory because it doesn't need to store slices in memory. I case of this you can render 3D fractals in very high resolution.  But... in your renders there is no shadows.
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David Makin
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« Reply #131 on: August 02, 2009, 08:50:12 PM »

Hi David
You made very good and accurate render. I'm very impressed by very high resolution of this render. Using your render algorithm you don't have to use large amount of memory because it doesn't need to store slices in memory. I case of this you can render 3D fractals in very high resolution.  But... in your renders there is no shadows.


Thanks. A disadvantage is that every view requires a complete re-render whereas once you have your 1500 jpgs you can more rapidly render from any viewpoint. Another disadvantage is that the method is quite prone to irregular aliasing effects which isn't so bad when using a regular grid as you are, I'm considering trying out a "snap to grid" option.

As to shadows, reflections, transparency/diffraction and better lighting (with something approaching radiosity) and the ability to render a complete scene rather than a single fractal - I'm working on a set of ray-tracing classes for UF5 but it will take some time especially as the simulated radiosity is something I've never tried before smiley
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« Reply #132 on: August 02, 2009, 08:56:59 PM »

congrats dave, this is simply awesome!
hehe, was it fun to code ? cheesy
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David Makin
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« Reply #133 on: August 02, 2009, 09:01:08 PM »

It was fun smiley
Just a little tricky with the final "tweaking" of the default "detail level" which I've set so it renders very quickly (around 4* faster than the render I uploaded) but without too many errors. So you can leave the detail level set at "1" while setting up and then increase it as necessary for the final render.
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« Reply #134 on: August 12, 2009, 12:26:40 AM »

I'm new to this discussion, but I've been emailing David Makin, twinbee, and lycium about 3D fractals and they've been helpful. I wrote a simple isosurface ray tracer to render these fractals with options for global illumination and participating media. Here are my results:
http://bugman123.com/Hypercomplex/index.html


* Mandelbrot-White2-large.jpg (202.23 KB, 1024x768 - viewed 1262 times.)

* Mandelbrot-White1-large.jpg (218.49 KB, 560x560 - viewed 1264 times.)

* Minibrot-Tricorn.jpg (27.17 KB, 280x280 - viewed 1198 times.)

* Minibrot-Quaternion.jpg (18.83 KB, 280x280 - viewed 1202 times.)
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