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Author Topic: how is whipped cream reflected in fractal dimension ?  (Read 2024 times)
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cKleinhuis
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« on: December 01, 2010, 11:17:32 PM »

hello all,

i am wondering about some fact concerning the mandelbulb and fractal dimension, so first the facts:

- the space filling curve of its surface posseses dimension 3 - this is because a limited interval is projected to an unlimited set of points


question 1: how are space filling curve dimensions related to fractal ( box-counting ) dimensions ?


so, but what about the whipped cream sections of the bulb ?


as far as i know a box counting analysis can only take part of the image, and would yield different results for different areas ( e.g. 0 for complete outside, between 2..3 for borders, 3 for inside ) , would it be right to state that in contrary to the 2d mandelbrot the mandelbulb posesses areas of non-fractal dimension on its border? e.g. if only a whipped cream area would be analysed it would result in a non fractal dimension ( 3 ) and for the borders which are "crumpled" it would yield something between 2..3 or 3..4 ?!??!

i am not sure about these facts, but i would like to make a box counting analysis of the fractals, to show what the problems of the mandelbulbs are that it is considered NOT to be the holy grail...
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« Reply #1 on: December 02, 2010, 01:10:34 AM »

Even if a tiny tiny volume of the set has dimension 3 then the whole set is dimension 3.
We know it is dimension 3 because the cross section at j=0 has dimension 2 so it must have dimension 3.
The whipped cream sections might also have dimension 3, since an extruded mandelbrot (a mandelbrot prism) is whipped cream and has dimension 3.

I am very confident that the mandelbulb has fractal dimension 3. I am also very confident that it is NOT a 'true' 3d mandelbrot, nor has Twinbee ever claimed it to be. There isn't a true 3d mandelbrot because there is provably no 3d analog of complex numbers.
It IS however an interesting and fascinating shape.
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cKleinhuis
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« Reply #2 on: December 02, 2010, 01:21:30 AM »

righty right, but how would you classify the "whipped" cream?
 it is surely not reflected in the fractal dimension, nor in space filling properties ...

the question is how to describe a "compound" object, consisting of euclidean and chaotic parts
 (@tglad pm me about how to prove the 3d analogy of complex numbers doesnt exist ( is it hamiltons problem ?))


does box counting give other results on a extruded mandelbrot than 3 ?
 
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Tglad
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« Reply #3 on: December 02, 2010, 04:33:56 AM »

"but how would you classify the "whipped" cream? it is surely not reflected in the fractal dimension"
Thats right, a surface can have fractal dimension 3 (as rough as is possible) and still be extruded 'whipped cream'.
To get a fractal that is fully rough in all directions I think you need to specify:
 the surface is dimension 3
 and
 all paths over the surface are dimension 2

I think it might be Hurwitz's theorum that there are only 4 algebras with properties like the reals and complex numbers: reals, complex, quaternions, octonions.
There is also that problem that dimensions>2 will not be conformal on any operation that acts like a ^2, due to Liouville's theorum.

Box counting should also give 3, since mandelbrot is not multifractal its hausdorrf, box counting dimensions should be equal, i.e. 2.
« Last Edit: December 02, 2010, 04:36:59 AM by Tglad » Logged
Prokofiev
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« Reply #4 on: December 02, 2010, 09:58:20 AM »

does box counting give other results on a extruded mandelbrot than 3 ?
The box-counting dimension is mathematically never smaller than the Hausdorff dimension (K. Falconner). The Hausdorff dimension being equal to the dimension of the space in which it lies, the box-counting dimension necessarily equals 3, also.  wink
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cKleinhuis
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« Reply #5 on: December 02, 2010, 11:18:49 AM »

hmm, so, if i see this right, the "whipped cream" sections of an object can not be described in such a way ?
is it just visually "seeable" ?

i mean, if we are continue searching for the "holy grail" we would need an analysis that is not satisfied with the current method,
at first glance my thought was, somehow the fractal or boxcounting dimension is not the same for all sections of its border, so,
if there is a section of the border with an integer dimension it could mean that this object is NOT the holy grail...

but as you described, this is not possible ... sad

any other ideas for classifying such objects ?
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Prokofiev
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« Reply #6 on: December 02, 2010, 01:13:22 PM »

but as you described, this is not possible ... sad

any other ideas for classifying such objects ?

As Tom said, we can define our "wipped cream" as a subset with Hausdorff dimension 3 but with paths having dimension 1 (or at least smaller than 2). That's a way to classify our objet as "3D-but-with-smooth-lines".

But the way, we have not formally proven, yet, that this is the case for our 3D mandelbulb ! Regarding fractals, we must never trust our eyes. A formal proof would be necessary, in my opinion. Can we prove our wipped cream is really as smooth as it looks like ? Remember that the boundary of the antenna of the Mandelbrot set has dimension 2, is spite of its aspect...
I am not talking about a numerical box-counting analysis because we know such analysis can yield very wrong results (and they certainly will, concerning the Mandelbulb).
« Last Edit: December 02, 2010, 01:48:04 PM by Prokofiev » Logged

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Alexis
cKleinhuis
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« Reply #7 on: December 02, 2010, 01:56:25 PM »

wouldnt box counting, e.g. only of the border of the "whipped cream" would give hints for being an euclidean object ?
perhaps it is easier to just use the box counting on a 2d slice, where the whipped cream border would look like a julia set with seed 0 ?
i mean, the fractal dimension is non-integer on every subset of the border of the mandelbrot, but this would not hold true
for the whipped cream sections of the bulb...
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Prokofiev
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« Reply #8 on: December 02, 2010, 03:28:40 PM »

wouldnt box counting, e.g. only of the border of the "whipped cream" would give hints for being an euclidean object ?
No. I'm affraid.

perhaps it is easier to just use the box counting on a 2d slice, where the whipped cream border would look like a julia set with seed 0 ?
I'm not sure what you mean, but yes, the fractal dimension of 2D slices can be relevant and useful, (although not entirely sufficient, on second thought: it does not detect the presence of non-planar smooth paths) .
« Last Edit: December 02, 2010, 03:30:49 PM by Prokofiev » Logged

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Alexis
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« Reply #9 on: December 02, 2010, 09:04:58 PM »

For a small enough volume, whipped cream will appear as an extrusion of a 2D fractal. Our eyes detect this quite well. One option to detect this would be to calculate a small 2d cross section (XY plane for example). Then take a small step orthogonal to the cross section (dz in the Z direction) and calculate another 2d cross section. The cross sections can be convolved to find the best displacement match (dx and dy). The resulting vector (dx,dy,dz) represents an estimation of the least fractal direction.
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