Title: What's the volume of the Mandelbox? Post by: greentexas on November 09, 2016, 12:07:28 AM Could someone figure out the volume of the Mandelbox? It could be that the volume of this fractal is zero (like the Menger sponge), and I've seen the Mandelbox emptying.
Title: Re: What's the volume of the Mandelbox? Post by: Tglad on November 10, 2016, 05:45:59 AM I think it is zero (i.e. its fractal dimension is less than 3)... however I don't have proof and it seems to get denser as you get towards the centre. So there might be a depth at which no points can escape.
Title: Re: What's the volume of the Mandelbox? Post by: KRAFTWERK on November 10, 2016, 07:49:10 PM Not bad to get an answer from Mr. Mandelbox himself.
And I love the end of the answer "So there might be a depth at which no points can escape." Sounds like a poetic description of a black hole... O0 Title: Re: What's the volume of the Mandelbox? Post by: Buddhi on November 10, 2016, 10:08:27 PM Here is some kind of experiment to support Tglad's answer. When maximum number of iterations is very high then Mandelbox starts to be empty. Theoretically when maxiter will be infinite number, then Mandelbox will disappear.
Mandelbox was rendered here as a cross section at z = 0 (http://nocache-nocookies.digitalgott.com/gallery/19/640_10_11_16_10_04_01.png) Title: Re: What's the volume of the Mandelbox? Post by: DarkBeam on November 10, 2016, 10:40:19 PM Nice idea Buddhi but I think it is not a way to demonstrate that fact, who knows if it is due to an artefact of the distance estimation?
The topic is very interesting of course :) Title: Re: What's the volume of the Mandelbox? Post by: Tglad on November 11, 2016, 01:25:54 AM Great images. Josleys also talked about the Mandelbox disappearing for higher iterations (http://images.math.cnrs.fr/Mandelbox.html?lang=fr (http://images.math.cnrs.fr/Mandelbox.html?lang=fr))
My thought on this is that having zero volume (or D<3) does not necessarily mean that the object should be invisible. A plane has zero volume and is visible, and any 'shell' fractal is still visible: (https://sites.google.com/site/simplextable/2-simplex/0100.jpg) and 'foam' fractals are still visible: (https://sites.google.com/site/simplextable/2-simplex/1110.jpg) For certain angles at least, it looks like Menger sponge and maybe even 3D cantor dust would be visible... I give an example here http://www.fractalforums.com/3d-fractal-generation/the-mandelbox-is-tenuous/msg15558/#msg15558 (http://www.fractalforums.com/3d-fractal-generation/the-mandelbox-is-tenuous/msg15558/#msg15558). For fractals that definitely are ultimately invisible (their 2D screen projection has fractal dimension < 2), such as trees, an interesting topic is that there are different ways to render it that keep it visible without it being an approximation http://tglad.blogspot.com.au/2012/12/dimension-aware-rasterising.html (http://tglad.blogspot.com.au/2012/12/dimension-aware-rasterising.html) (I even made a paper on it 'Improving image clarity using local feature dimension'), but it has never been attempted in 3D. Title: Re: What's the volume of the Mandelbox? Post by: Buddhi on November 11, 2016, 07:24:59 AM You are right about disappearing. The fractal will be still visible but all remaining surfaces will have zero thickness. That's why they disappear when is used used raymarching with bisection algorithm.
Title: Re: What's the volume of the Mandelbox? Post by: woronoi on November 11, 2016, 07:33:54 AM Cantor dust (https://en.wikipedia.org/wiki/Cantor_set#Cantor_dust). That was a great answer for me.
Title: Re: What's the volume of the Mandelbox? Post by: gurroa on November 11, 2016, 10:52:45 AM I am counting virtual size of 3D fractals from the lowest possible rendered resolution. Program FractaloScop can get to 0.0000001. If we imagine this structural resolution as a 1 cm, 1.0 represents 100km, 2PI's MandelBulb is virtualy a 628km wide ball. If we imagine it instead as a 0.1 mm, 1.0 still represents 1km and MandelBulb shrinks to a 6.28km ball. And so on... The same logic can be applied to MandelBox too, but there is a catch. MandelBox can be easily resized and so heavily vary in size. I have changed almost each line of equation (just to get more readable inner space) and the default size for my program is now set to 12. So the mandelbox for me is a virtual box sized 1200km when the smallest geometry detail rendered as a pixelized block is a 1cm. Maybe you can use the same logic to count virtual volume too. This video is showing what I mean by "unreadable inner space" in almost :-) default MandelBox. I have called it the Abstract shortcut. https://www.youtube.com/watch?v=60SQel4zsr0 After my changes the current inner space become more like this at only really few iterations. https://www.youtube.com/watch?v=NWM15uJYI9A |