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Author Topic: The Mandelbox is tenuous..  (Read 6915 times)
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JosLeys
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« on: March 31, 2010, 11:35:09 PM »

From all the published images, it seems that the Mandelbox is a solid object. If we slice through it, we see complicated patterns of cavities, but they seem to be surrounded by solid material.
My point is that all this is not true at higher iterations.

The inside of the classic Mandelbrot set is "solid". Any point in any area of the inside of the set will stay bounded, even at an infinite number of iterations. For the Mandelbox , this is not the case.

The film below shows a series of 100 slices through the box, so from z=-6 to 6, with iterations set at 13.
http://www.josleys.com/gfx/Mandelbox2D-04.mov  (18Mb, be patient)
You will recognise familiar patterns of the "standard" box ( scale=2)

Now the same 100 slices at 10.000 iterations:
http://www.josleys.com/gfx/Mandelbox2D-05.mov (277kb, this will load quickly)

There now only appear to be 'inside' points between -3 and 3. ( with, strangely enough, a set of points at -6 and +6)

In none of the slices do there appear to be contiguous areas of inside points. We see nothing but collections of 'dust'.
It is to be noted also that without the inversion step, the limit set becomes a cube half as large as the 'box' (vertices at <+/-3,+/-3,+/-3> instead of  <+/-6,+/-6,+/-6>) (see the image below : the area between 3 and 6 gets more and more tenuous: the spacing between the lines appears to be Cantor dust)

It would be nice if somebody could confirm these results. A voxel render at high iterations should confirm this.


* Mandelbox2D-01.jpg (94.7 KB, 640x640 - viewed 321 times.)
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Tglad
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« Reply #1 on: April 01, 2010, 03:38:08 AM »

The second mov doesn't run for me, but it sounds wrong, what is your bailout distance? I think it needs to be at least 12 (144 in UF), I set it to 1024 to be safe.
Whole sections shouldn't disappear with higher iterations, however, I think you are right that any surface that appears solid will probably have small cracks and gaps and arches in it if you zoom close enough.
« Last Edit: April 01, 2010, 03:44:35 AM by Tglad » Logged
JosLeys
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« Reply #2 on: April 01, 2010, 08:25:25 AM »

Both films run without problems from here.

The bailout was 15, but I get the same result with a bailout of 15.000..
Note that there is a huge difference between a surface with pits and cracks, and a cloud of dust..
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JosLeys
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« Reply #3 on: April 01, 2010, 11:05:39 AM »

I took Dave Makin's UF implementation, which has an option to demand  a minimum number of iterations before drawing something solid.
In the images below, the first has no such demand, and the second demands a minimum of 100 iterations.
You see that of the original cube, there remains something dusty, at exactly half the size..


* Mandelbox 0its.jpg (214.22 KB, 600x600 - viewed 467 times.)

* Mandelbox 100its.jpg (62.01 KB, 600x600 - viewed 337 times.)
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Tglad
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« Reply #4 on: April 01, 2010, 11:39:49 AM »

I just don't have quicktime smiley
I think its a bug in the minimum iterations option on Mandelbox. But did the videos use this min iterations?
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JosLeys
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« Reply #5 on: April 01, 2010, 12:10:12 PM »

The videos us a specially written program in UF to show only 2D, so no distance estimation is involved. I take a pixel and iterate it through the folding and inversion. If it does not bailout it turns blue, if not, white..
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JosLeys
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« Reply #6 on: April 01, 2010, 12:27:14 PM »

Just to add that my remarks about the Mandelbox are not meant to be disparaging in any way. I love the pictures!
I'm just trying to dissect the animal mathematically.
What I think is that the Mandelbox has properties like the Cantor set : the higher the iterations, the more dust..

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David Makin
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« Reply #7 on: April 01, 2010, 01:11:23 PM »

To be honest just looking at the transforms that make up the Mandelbox formula I would expect it to be made up of Cantor Dust.
As to the appearance of renders using my formula but with a high "min. iterations" value, I think the result is more of an issue of very small points being missed by the render - i.e. it is Cantor Dust but there should be a lot more of it.
The "Min. iterations" option was only really designed to skip small iteration counts for on Mandelboxes where there are discontinuities in the smoothness of the smoth iteration colouring - here for example a value of say 8.5 is immediately followed by one of say 8.4 which looks to my renderer as if we've hit solid but the point is still "outside" and the driop in value is simply due to a discontinuity in the smooth iteration values, actually the option is obsolete if you enable "Double check for solid".
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JosLeys
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« Reply #8 on: April 01, 2010, 03:56:23 PM »

Agree, but "seeing" the dust particles becomes virtually impossible at high iterations because they become smaller and smaller and smaller...
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Timeroot
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« Reply #9 on: April 01, 2010, 09:47:11 PM »

A while ago, I too had played with these slices. The only things I remember was how at z=0, z=+/-2, z=+/-4, you get a "full" slice; something that appears to fill a whole plane. On the other hand, z=2.00001, with sufficient iterations, gave cantor dust. I assume this is due to some very fragile equilibrium between the folding and spherical inversion; I am curious what would happen if the folding and inversions are tested with < and <=, since these might "break" that one slice.

I do not remember any unusual dust, though. Do you think you could try rendering a video of any random slice (say, z=2.3 or whatever) with increasing iterations? This might show us how the differences came about.

Interesting find, for sure!  Azn
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JosLeys
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« Reply #10 on: April 01, 2010, 11:27:52 PM »

To Timeroot:

See below the slice at z=2.3 at 15,20,35 and 75 iterations.

A solid square is to be found at z=+/-1, +/-3, and +/-5, even at high iterations.(and of course also at the same values for x and y, due to symmetry)


* Mandelbox2D-06_15.jpg (125.27 KB, 480x480 - viewed 784 times.)

* Mandelbox2D-06_20.jpg (125.76 KB, 480x480 - viewed 812 times.)

* Mandelbox2D-06_35.jpg (89.69 KB, 480x480 - viewed 770 times.)

* Mandelbox2D-06_75.jpg (99.97 KB, 640x640 - viewed 301 times.)
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Tglad
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« Reply #11 on: April 02, 2010, 12:40:59 AM »

JosLeys I think you're on to something here. It does make sense that it would be like dense Cantor dust.
Thinking about it, even if the cracks are tiny each iteration, you still end up with 0 volume... if you have a line and you remove 1% as the biggest crack.. then all the remaining percent segments also have tiny cracks 1% of their own length, then the total line length is .99 * .99.. continuing you get .99^infinity which is 0 total length.
You get the same result with a Sierpinski tetrahedron and with a Menger sponge, and the renders would also disappear as sparse dust because you are rendering with a probe that is very thin compared to the pixel size.

A Menger sponge or carpet renders better if the probe is about the width of each pixel, ie colour it black if you get within half a pixel radius of an inside point.
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Timeroot
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« Reply #12 on: April 02, 2010, 12:50:09 AM »

I agree with Tglad, it seems that the fractal is still "correct", but it becomes so sparse that nothing happens to hit upon some of the dust. I'm sure that if you rendered it in 2D with distance estimation, it would stay the same regardless of the iterations. (Even with high minimum iterations).
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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
JosLeys
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« Reply #13 on: April 02, 2010, 12:54:57 AM »

..hmm
If a fractal turns to dust, distance estimation is not a good method I think..
My 2D images are made by iterating the pixel through the folding and inversion , and either it bails out or it does not.
If it bails out then we have empty space on that spot..
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Timeroot
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« Reply #14 on: April 02, 2010, 01:35:59 AM »

but just whether or not it is on that spot is not a good indicator of whether the fractal is actually "there". Distance estimation can tell you, "is there fractal anywhere within this pixel?", and if so, color it for you. If you iterate a whole buncha times, Almost All points will be excluded, and so Almost No points would be plotted on screen. Try look at the Mandelbrot set. It's not even dust. But unless you're looking at some bulb or some minibrot, you won't really get to see any inside points. The high iteration value converging there just tell you there's a point right in the middle; but chances are none of the pixels landed exactly on the scraggly little line.
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Someday, man will understand primary theory; how every aspect of our universe has come about. Then we will describe all of physics, build a complete understanding of genetic engineering, catalog all planets, and find intelligent life. And then we'll just puzzle over fractals for eternity.
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