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Welcome to Fractal Forums > Fractal Math, Chaos Theory & Research > General Discussion > Hausdorff duminsion=fractal dimension?
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Author Topic: Hausdorff duminsion=fractal dimension?  (Read 2851 times)
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luuc
Guest
« on: December 08, 2006, 12:08:44 PM »
I've been searching already the find an awnser to my question; whether a Hausdorff dimension is the same as a fractal dimension.

On the Wikipedia page of the Hausdorff dimension (http://en.wikipedia.org/wiki/Hausdorff_dimension) there is written:

Quote
that is a number in the closed infinite interval [0, ∞])

Does this mean the dimension goes to an infinite number?
I thought the (fractal) dimension was from 0 till 3, and no further.
Maybe I don't understand, but I hope someone can help my with this question.

Thanks.
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gandreas
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« Reply #1 on: December 08, 2006, 05:25:14 PM »
Short answer - a Hausdorff dimension can be the same as a fractal dimension.

Longer answer:

From http://mathworld.wolfram.com/HausdorffDimension.html:
Quote
In many cases, the Hausdorff dimension correctly describes the correction term for a resonator with fractal perimeter in Lorentz's conjecture. However, in general, the proper dimension to use turns out to be the Minkowski-Bouligand dimension (Schroeder 1991).

and from http://mathworld.wolfram.com/FractalDimension.html:
Quote
However, it can more generally refer to any of the dimensions commonly used to characterize fractals (e.g., capacity dimension, correlation dimension, information dimension, Lyapunov dimension, Minkowski-Bouligand dimension).

So basically, "fractal dimension" isn't rigorously defined, but has multiple common uses, one of which corresponds to the same value as the Hausdorff dimension.  Sort of likes saying "how big is something" can be "about the size of a golf ball", "1.5 inch sphere", "4 ounces", "fits in the palm of your hand", etc...
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himalayanjava
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« Reply #2 on: March 16, 2007, 08:49:39 PM »
I thought the (fractal) dimension was from 0 till 3, and no further.

Is he right? What's the probable value range for FD(Fractal Dimension)?
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himalayanjava
Guest
« Reply #3 on: March 17, 2007, 11:08:30 PM »
No replies? what does it mean? There is no CONSENSUS on "Fractal Dimension" value range?
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lkmitch
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Posts: 238
« Reply #4 on: March 22, 2007, 08:22:04 PM »
The probable range of a fractal dimension is about the same range as the topological dimension.  That is, for fractals with which a typical person might come into contact, 0 - 4.  But, there could be hyper-dimensional fractals, with fractal dimensions unbounded.

Kerry
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himalayanjava
Guest
« Reply #5 on: March 24, 2007, 04:30:10 PM »
thanks for making the things clearer, it will certainly help to get clearer view on the matter for those like me who r just entered in this field.

And, it will be again helpful if you can provide some information about CALCULATION of Fractal Dimension. I have heard there are varies ways, methods and formulas. What's difference between Box Counting and Correlation Dimension calcuation?
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himalayanjava
Guest
« Reply #6 on: March 26, 2007, 05:48:56 AM »
Thank you for the link. as you have said, one PICTURE is more than thousand word.

If there you found something about Corretion, don't foreget to share.  Thank you once again.
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