Hausdorff dimension of the Mandelbulb

[Prokofiev]

Hello,
I am a new member in fractalforums, but not quite new in fracal geometry.
I am the guy who commited this : http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension, well, most of it. Among other things.
I followed the fascinating efforts towards the discovery of the mandelbulb last year.
The Mandelbulb certainly deserves its place in this list, but where ?

On first thought, its fractal dimension should be 3, very probably, since it adds a full dimension to the boundary of the Mandelbrot set, which has already dimension 2 (Mitsuhiro Shishikura 1991). And this is certainly true for all powers.

Has anyone studied the subject seriously ? Or at least calculated its box-counting dimension ?

Thanks for your help !

[ker2x]

I have the excellent book by benoit mandelbrot "Les objets fractals. forme, hasard et dimension." which describe clearly the non-integer fractal dimensions.
It's odd that i never seen any topic about it here.

i'm glad you wrote one.
Unfortunally, i can't help.

[Sockratease]

Hello and Welcome to the forum!

Glad you joined in rather than just lurking.

But sorry, I too am unable to answer your question.  However, there are folks here who surely can.  Give them time to spot this thread...

[msltoe]

One of the two Mandelbulb inventors, twinbee, did some preliminary box counting on his web page:
http://www.skytopia.com/project/fractal/2mandelbulb.html#stats
He sees a value of ~2.945 for the 8th order Mandelbulb. If you know of a formal method of calculating the Hausdorff dimension, you may be to able modify the source code of a program such as Mandelbulber by Buddhi.
We also have another intriguing 3-D fractal called the Mandelbox that could be analyzed.

-mike

[kram1032]

I guess, K-IFS change their dimensionality, huh?
Because their Hausdorff Dimension would be interesting too

[Prokofiev]

He sees a value of ~2.945 for the 8th order Mandelbulb.
-mike

Thanks Mike, but that was Daniel White's estimation of the volume, if I read correctly.
And, as he adds just underneath, he has "no idea about its Hausdorff or fractal dimension". So far ?

I do agree the Mandelbox should also be studied.

[Nahee_Enterprises]

    Hello,   I am a new member in FractalForums, but not quite new in fracal geometry.
    The Mandelbulb certainly deserves its place in this list, but where ?
    Has anyone studied the subject seriously ?  Or at least calculated its box-counting dimension ?

Greetings, and Welcome to this particular Forum !!!   

Good questions, but not something I can answer for you nor point to a location to acquire the answers.  Most likely there are none at present.  Sounds like a good project that you might wish to take on, if you so desire.   
 

[Calcyman]

I assume everyone is discussing the boundary of the Mandelbulb here -- the boundary is fractal, whereas the interior is solid, with a dimension of exactly 3.

The boundary of the ordinary Mandelbrot set has a Hausdorff dimension of 2 (and thus could even have an area!). The Mandelbulb does not look convoluted enough to have a Hausdorff dimension of 3 -- if it falls short, then it is definitely not the ideal 3D extrapolation of the Mandelbrot Set.

Because the triplex algebra is horribly complex (no pun intended!) to analyse, determining the fractal dimension analytically may be intractable.


If I remember correctly, the Mandelbulb is multiply connected (it is topologically inequivalent to a 3-ball). In fact, coupled with its quasi-self-similarity, the Mandelbulb is genus-infinity, and therefore has an infinite Euler characteristic. An ideal 3D Mandelbrot Set should be genus-0.


The Mandelbox is another thing entirely. Again, assuming I have interpreted it correctly, it doesn't use an underlying 'algebra', but instead the fractal is determined by the bounding planes. A Mandel-dodecahedron should be equally possible, or indeed any convex polyhedron.



Are there any academic papers relating to these new objects, or the triplex algebra itself?

[Tglad]

Hi Prokofiev, and excellent work on that wikipaedia page I have looked at it often, its a great reference.
I can't give a value for the hausdorff dimension of either of these two objects but I am almost certain that they both are 'multi-fractals' so their hausdorff, box-counting dimension etc probably aren't the same, and the hausdorff dimension itself I think would change depending on where you look on the object... but maybe an overall average can be calculated.

The reason why I think they are multi-fractal is because the scale-factor in the iteraction varies with location, e.g. larger near the poles on the mandelbulb and on the mandelbox it is the ball-fold that gives varying scale factor.
For this reason I think that hausdorff dimension for the kaleidoscopic fractals should be much easier to calculate, they aren't multi-fractal.

By the way, I know that article is on Hausdorff dimension, but it would be also cool to know the 'size' of each of those fractals, e.g. is the unit moore curve 'larger' than the unit peano curve or Hilbert curve?
Hausdorff dimension is a bit like saying a disk area is order(r^2), it would be nice to know that it is in fact pi*r^2, here's a ref: http://www.fractalforums.com/mathematics/how-long-is-the-coastline-of-great-britain/

[Calcyman]

By the way, I know that article is on Hausdorff dimension, but it would be also cool to know the 'size' of each of those fractals

One measure I have devised (probably re-invented) is as follows:

* Divide the space into hypercubes of side length x units.
* Count the number of hypercubes that contain at least one point of the set (call this value n), and calculate n*x^D, for some value of D.
* As x becomes arbitrarily small, n*x^D will either approach 0 (if D is greater than the Hausdorff dimension), infinity (if D is less than the Hausdorff dimension), or a finite value (if D is the Hausdorff dimension).
* This finite value, the limit of n*x^D as x --> 0, is the size of the set, in units^D. So, for example, this could be used to calculate the size of a Sierpinski triangle, in metre^(ln(3)/ln(2)).

[kram1032]

would it have any sense to do those calculations over the whole parameter space? (Which of course would be way more than 3D)
Maybe even the generalized multifractal dimension and lacunarity. It could end up to somehow highlight nice parameter sets

[Paolo Bonzini]

Are there any academic papers relating to these new objects, or the triplex algebra itself?

I would not call it academic at all, but there's

http://github.com/bonzini/mbulb/raw/master/mbulb.pdf

Paolo

[Tglad]

Calcyman, that sounds right.
 My own guess is that they both have an average hausdorff dimension of around 2.8. And yes, this is the dimension of the mandelbulb surface, but the dimension of the whole mandelbox.

[Prokofiev]

Hello,
Interesting comments.
Apparently nobody has attempted this calculation.
Calcyman, I'm affraid your measure will always give 1 for any self-similar fractal pattern of side 1 (at least for strictly self-similar patterns such as the Sierpinski triangle, your definition derives from the box-counting dimension).
Now, a point to keep in mind : The Hausdorff Dimension of a set is the highest dimension encountered in any of its subsets (Falconer, Mandebrot...). There is no such thing as an "average Hausdorff dimension". The boundary of the Mandelbulb, as a whole, "seems" to have a dimension around 2.7 - 2.8, but there could be somewhere a subset, no matter how small, more dense, with a dimension closer to 3. (The boundary of the Mandelbrot set has dimension 2, in spite of what we could guess, because of such subsets).
I agree it must be really difficult to get an analytic result of the Hausdorff dimension with such a difficult algebric complexity. I would not know where to start.
At least we could try to look for a "dense-looking subset" and measure its box-counting dimension. That would give a reasonable lower bound.
Actually we should search in the neighbourhood of the xy-plane, the highly dense regions of the corresponding Mandelbrot Set.

[Tglad]

I'm not sure that is right about the mandelbrot set, the boundary looks equally rough everywhere to me, and I'd be surprised if you can find a subset of the boundary with a dimension < 2.
I also don't think it is right to say it will give 1 for a pattern of side 1, for example, a Koch curve (from a to b) has side 1, but a double koch curve (from a to be then back) is twice as much, at every stage of box counting for instance there will be twice as many boxes filled with the curve. In fact I looked it up and its called the Hausdorff measure of the fractal.