|
JosLeys
|
 |
« Reply #15 on: April 02, 2010, 09:17:03 AM » |
|
OK, I see what you mean, but if the fractal has turned to disconnected specks of very, very small particles, then I'm not sure that distance estimation will tell us more. The Mandelbrot set is connected and distance estimation allows to find the fine tendrils between minibrots, but I think this is different..
|
|
|
|
|
Logged
|
|
|
|
|
Timeroot
|
|
« Reply #16 on: April 02, 2010, 10:26:24 PM » |
|
But it's not different! Look at the Julia set (-1.5, 0.  - it has an infinite number of points. Chances are, not a single pixel on your screen is "inside". On the other hand, the iteration values - or Distance Estimation, if you prefer, tell us where the points are, even if we don't land on them. |
|
|
|
|
Logged
|
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.
|
|
|
Tglad
Fractal Molossus
Posts: 703
|
|
« Reply #17 on: April 07, 2010, 02:21:50 AM » |
|
Timeroot is quite right I think, but I think it is great JosLeys that you are looking into what this shape is. I think your post has lead us to a collective conjecture that the Mandelbox is nowhere dense, so has a volume of 0. I wouldn't even like to think how that could be proved though  And such a fractal does have some weird properties, as you saw 0 radius probes will miss the set 100% of the time with enough iterations. That doesn't mean that there is nothing there, it would miss the edge of a circle 100% of the time too. It means we need to be careful how to render a 2d Mandelbox... one problem with just colouring black if there exists a point inside the radius is that there is no anti-aliasing, but too much anti-aliasing and the points will fade away again, a similar problem to rendering the smooth edge of a circle. It is actually easier in 3d, strangely a 0 width ray of light will bounce off the Mandelbox even though it has no volume, it can be demonstrated on small-hole cantor dust. So should always look solid/opaque. I guess another question about this object, what do you measure as the fractal? Is it the surface area that is fractal, or is it just the set of points that never escape? What is its topological dimension I wonder. |
|
|
|
|
Logged
|
|
|
|
|
Timeroot
|
|
« Reply #18 on: April 07, 2010, 02:33:41 AM » |
|
I believe that, because it is nowhere dense, it's surface is the whole fractal (that is, the set of points that don't escape). And even if that's false, the fractal would be considered the points that don't escape. We say the Mandelbrot Set is even the fatty insides, even though only the border is truly fractal. I think that, even in 3D, rays would generally miss the solid. DE is implemented. .... - No, wait - there are actually the full 2D slices which are always reoccurring, giving it something to hit. But with ray tracing, there's the chance that it would skip right over that very thin slice, so, yeah, I guess you actually would need DE to see anything.  |
|
|
|
|
Logged
|
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.
|
|
|
|
KRAFTWERK
|
|
« Reply #19 on: April 07, 2010, 09:12:53 AM » |
|
... as lead us to a collective conjecture that the Mandelbox is nowhere dense, so has a volume of 0. I wouldn't even like to think how that could be proved though Make a 3D print of it?  It would be pretty cheap if you are right about zero volume  |
|
|
|
|
Logged
|
|
|
|
|
kram1032
|
|
« Reply #20 on: April 07, 2010, 06:21:22 PM » |
|
cheap and unstable  |
|
|
|
|
Logged
|
|
|
|
|
reesej2
Guest
|
|
« Reply #21 on: April 09, 2010, 08:52:21 AM » |
|
I believe that, because it is nowhere dense, it's surface is the whole fractal (that is, the set of points that don't escape).
Well, if it's nowhere dense it's kind of hard to say that it has a "surface" at all... But when it comes to "seeing" the fractal, why not just use rays with nonzero width? A zero-radius ray is almost guaranteed to miss, if it's nowhere dense, but a "thick" ray would hit whenever the zero-radius ray would have come close. It has a physical analog, I guess, since light beams do have thickness... Anyway, distance estimate would help hugely with actually displaying that. Question: The hypothesis that the Mandelbox is nowhere dense entails that, as the iteration count gets higher and higher, the image should gradually vanish (as long as your rays have zero width). Has anyone actually noticed this? Or does it just exhibit the "solidifying" behavior of the Mandelbrot or Mandelbulb? |
|
|
|
|
Logged
|
|
|
|
Tglad
Fractal Molossus
Posts: 703
|
|
« Reply #22 on: April 09, 2010, 09:20:21 AM » |
|
"Well, if it's nowhere dense it's kind of hard to say that it has a "surface" at all.." True, it has no boundary between an inside volume and the outside.. it might be better to think of the Mandelbox as being just a set of points in space. In 2d the box will vanish with higher and higher resolution, just like a circle will vanish with higher and higher resolution. But just as a sphere shell is visible regardless of the resolution (despite it having no volume and being nowhere dense) I'm most certain that the 3d Mandelbox will not vapourise at higher resolutions. It is very similar to cantor dust, and this picture shows how diagonal light rays (red) hitting cantor dust do not penetrate regardless of the resolution... the maximum penetration just gets closer and closer to reaching the middle. 0 volume but 100% opacity. But as Timeroot said, in practice the 3d box might start to vanish as the rays will accidentally skip over the inside points.. so non-zero ray width would help with this I expect |
|
|
« Last Edit: April 09, 2010, 09:28:45 AM by Tglad »
|
Logged
|
|
|
|
|
matsoljare
|
|
« Reply #23 on: April 09, 2010, 11:04:56 PM » |
|
That's only true when it's from that exact angle, though....
|
|
|
|
|
Logged
|
|
|
|
Tglad
Fractal Molossus
Posts: 703
|
|
« Reply #24 on: April 10, 2010, 01:34:53 AM » |
|
It only starts to penetrate when the box angle gets closer to horizontal... and the smaller the gap in the dust the closer to horizontal it can get without penetrating. The Mandelbox has very small gaps compared to standard cantor dust. Additionally the sphere inversion mixes up the crack orientations so gaps won't ever line up sufficiently allow this sort of penetration.
Not a bulletproof arguement but its enough to convince me that the Mandelbox would look much the same with 0 width rays (or infinite resolution) as it does with the renders we do now.
I think I've been thinking about fractals too much, my girlfriend heard me talk in my sleep yesterday, mumbling "its not very fractally" !
|
|
|
|
|
Logged
|
|
|
|
|
reesej2
Guest
|
|
« Reply #25 on: April 10, 2010, 08:50:14 AM » |
|
I'm convinced. Admittedly for certain angles, light would pass through perfectly, but an observer never views every point with the same angle. So the Mandelbox might look "thin" in some places but I don't think it would ever disappear. Regardless, though, the density poses a problem for rendering, because if these isolated points are missed in the steps (almost guaranteed without some sort of dynamic step size) then the Box WOULD disappear. [q]I think I've been thinking about fractals too much, my girlfriend heard me talk in my sleep yesterday, mumbling "its not very fractally" ![/q]  Know the feeling... |
|
|
|
|
Logged
|
|
|
|
|
visual.bermarte
|
|
« Reply #26 on: April 10, 2010, 01:13:01 PM » |
|
Hi, here you can see a test showing mandelbox's slices. A part of the same data set was used to create a .dxf file (with low-iterations). I used mandelbulb3D and imageJ; using higher iterations creates more problems with the isosurface (maybe because I'm using a 32 bit system).
<![CDATA[<a href="http://www.youtube.com/v/MDVhQJLPIm8&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/MDVhQJLPIm8&rel=1&fs=1&hd=1</a>]]> |
|
|
|
|
Logged
|
|
|
|
Jesse
Download Section
Fractal Schemer
Posts: 1013
|
|
« Reply #27 on: October 30, 2010, 06:44:58 PM » |
|
From what i have seen in renderings with high iterations, it looked like the scale 2 and minR 0.5 box just
is no more solid, but if you decrease the scale and/or increase the minR value, it "looked" like getting more
and more solid in certain areas.
I cant proof this mathematical yet, but looking at the case for the vectorlength to be between minR and 1,
the spherical folding becomes scale/vectorlength, what leads to nonlinear gradients with the lowest value at 1,
beeing -1?
If this gradient is getting below 1, there might be possible areas that dont get stretched... but these are only
some first thoughts of what i discovered so far.
Edit:
Duh, i think i made a mistake in the scaling of the graphic for my tests, it seems that the gradient does not
goes below the scale value!
That does not affects the conclusion from what i have seen with high iteration counts, but i am totally out of
doing an analytical explanation.
|
|
|
|
« Last Edit: October 31, 2010, 11:21:54 AM by Jesse »
|
Logged
|
|
|
|
|
September 08, 2010, 03:24:57 PM
Dilber MC Theme by HarzeM