Talk:Hausdorff dimension

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[edit] Talk 2003 -2006

"If M is a metric space, and d > 0 is a real number, then the d-dimensional Hausdorff measure Hd(M) is defined to be the infimum of all m > 0 such that for all r > 0, M can be covered by countably many closed sets of diameter < r and the sum of the d-th powers of these diameters is less than or equal to m."

I read infimium, so I'm clear on that, but this definition is still opaque to me. Please clarify. --BlackGriffen

The box-counting dimension is not the same thing as the Housdorff dimension! -- Miguel

Thanks, I changed that. Now all we need is an article on the box-counting dimension... :) -- Schnee 23:22, 13 Aug 2003 (UTC)

Hausdorff dimension isn't the only fractal dimension...it should have its own entry, and fractal dimension to talk about fractal dimensions collectively, it seems.

Done. -- Jheald 22:01, 19 January 2006 (UTC).

[edit] Banner

The banner is ridiculous and I am going to remove it. CSTAR 17:43, 25 Nov 2004 (UTC)

[edit] Doubling property

I put in a condition at the beginning which is basically the doubling property (which is of course much weaker). I'm not even sure I stated it correctly. Will the experts object? I perhaps should do my homework and consult my copy of Gromov-Semmes et-al. But hey, I'll let Tosha do that.CSTAR 17:49, 26 Nov 2004 (UTC)

Haha, ironic for the page-doubling bug to strike here. X) --[[User:Eequor|η♀υωρ]] 21:24, 26 Nov 2004 (UTC)

[edit] Name change

It's called Hausdorff dimension. Policy is that usual names should be used. This page should be moved back. Charles Matthews 19:31, 26 Nov 2004 (UTC)

It's moved back. Do you happen to know what the point of the name change was? CSTAR 19:43, 26 Nov 2004 (UTC)

It's User:Eequor on a stampede. See Besicovitch. Charles Matthews 20:08, 26 Nov 2004 (UTC)

I see; so because the two of you agree with each other, you must be correct? --[[User:Eequor|η♀υωρ]] 20:39, 26 Nov 2004 (UTC)
Charles Matthews and I do agree, and either we're correct or we're not. Whether we're correct or not ideally should be determined by a discussion of the relevant linguistic facts in this case (as I've tried to do below). If no consensus emerges, than some other mediation procedure is needed. That mediation procedure may reduce to some less desirable criteria for selection of the ultimate name.
Please note that neither of us is suggesting Besicovitch not be mentioned. BTW, Besicovitch's contribution in this area is in obtaining fairly precise bounds on the growth properties of coverings. Besicovith wrote all his papers mre than 20 years after Felix Hausdorff introduced Hausdorff measure. I don't see how you can even argue fairness here! CSTAR 21:13, 26 Nov 2004 (UTC)
Ah, that helps. Do you happen to have the dates of their publications, or where they were published? It would be nice to have the details of Besicovitch's contribution. --[[User:Eequor|η♀υωρ]] 21:21, 26 Nov 2004 (UTC)
Besicovitch indeed has a long list (in the 100s) of reputable publications, which is much longer than Haudorff's (although as I said, came much later than Hausdorff). Partly because Besicovitch lived much longer than Hausdorff he was much more prolific. Probably the most relevant one is
  • A. S. Besicovitch and H. D. Ursell, Sets of Fractional Dimensions, 1937, Journal of the London Mathematical Society, v12.
However, I freely admit I am not a historian of mathematics, so among the 100s of papers he wrote there may be other more relevant ones. The main point is that Besicovitch was a very accomplished analyst who developed many powerful techniques to compute many things, including Hausdorff measures. So I don't want to err in the other direction and not give him due credit.CSTAR 21:40, 26 Nov 2004 (UTC)

[edit] Historical accuracy

If you really want to be accurate, you should say something like

Though most authors use the term Hausdorff dimension, and indeed Hausdorff was the first to explicitly introduce this concept, some authors (notably Mandelbrot) also refer to it as Hausdorff-Besicovitch dimension, particularly since Besicovotch developed many techniques for determining sizes of coverings of sets, which in turn are useful for calculating Hausdorff measures of highly irregular sets.

ALso Hausdorff dimension is not a measurement and it also would be misleading to call it a measure. It's just a number assigned to a set. CSTAR 22:07, 26 Nov 2004 (UTC)

[edit] Hausdorff dimension

Why the name change? Virtually every authoritative work on the subject refers to the concept as Hausdorff dimension. For example

  • H. Federer Geometric Masure Theory, Springer -Verlag, 1969
  • K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985
  • M. Gromov with M. Katz, P. Pansu and S. Semmes, Metric Structures for Riemannian and Non-Riemannian Surfaces, Birkauser, 2001

CSTAR 19:56, 26 Nov 2004 (UTC)

On the other hand, Mandelbrot uses Hausdorff-Besicovitch dimension. It would be unfair to credit Felix Hausdorff but not Abram Samoilovitch Besicovitch, wouldn't it? --[[User:Eequor|η♀υωρ]] 20:01, 26 Nov 2004 (UTC)

Firstly, Mandelbrot is much more of a publicist than a serious mathematician. I'm not prepared to take that book as an authority on mathematics. We use fractal the way he does (for better or for worse), because it's his term.

Second point: Wikipedia naming policy is (generally) to use the most usual name. The article can discuss credit where due; the title should be the normal way of naming anything. Charles Matthews 20:19, 26 Nov 2004 (UTC)

Mandelbrot I'm afraid is much in the minority here. The list of sources calling it Hausdorff dimension is really very long. In addition to the three I mentioned above, one can also add:
* W. Hurewicz and Wallman, Dimension Theory, Princeton Univrsity Press, 1941
* F. Morgan Geometric Meaasure Theory, Academic Press, 1988
Whether or not it's unfair to give it a name is an entirely different matter, and is not Wikipedia's responsibility to assign names based on fairness or any other criteria. If you want to do a historical study to determine who invented the concept, then by all means do so. I will support your writing such an article and it will be useful. However, in this instance the overwhelming evidence is that it's called Hausdorff dimension by authorities in the area. The Wikipedia policy on the matter is quite clear.CSTAR 20:22, 26 Nov 2004 (UTC)
I am aware that the metric is often referred to as the Hausdorff dimension; however, to dismiss one researcher because "everybody else does" is POV. --[[User:Eequor|η♀υωρ]] 20:33, 26 Nov 2004 (UTC)
The discussion is in relation to how something, in this case a technical concept, is named. Concepts often are assigned names in a haphazard way, completely unrelated to who may have first discovered them. Buy using a name, no one is making a determination about credit. Bezout's theorem is one example. It is almost universally agreed that he had little to do with it.
Wikipedia should not be in the business of assigning names or creating extensions or modification of existing language (except in very limited cases, such as providing an auxiliary explanatory or as a temporary notational artifice). Moreover, I have presented many academically respected references showing that indeed "everybody else" (other than Mandelbrot) assigns this name to the dimensionality concept in question. How is relying on the fact that "everybody else" (other than Mandelbrot) uses a particular name an insatnce of POV? Language is a social phenomenon. It's the collective result of everybody's behavior.
It may unfair but that's the way linguistic behavior works.CSTAR 20:57, 26 Nov 2004 (UTC)
Names are area where "absolute NPOV" is hardly possible. Be it a mathemathical concept, Polish city or anything where various POVs may exist. Wikipedia policy to use the name the reader would most propably use is reasonable. Frequency of usage is decidable and factual. Historical justice is not. Best what can be done about NPOV is to note various names in the article and create redirects. (See Prim-Jarnik's algorithm) --Wikimol 22:31, 26 Nov 2004 (UTC)
That's right - mathematical names are full of incorrect history. The history is properly in the article, not the title. The argument offered does not, in my view, have much to recommend it. Charles Matthews 22:39, 26 Nov 2004 (UTC)
I agree with Charles and CSTAR and I moved it back -Lethe | Talk
I think it should be Hausdorff dimension. That's the term commonly used. As such it is the appropriate title. Whether justified or fair, most people call it this, and Wikipedia policy is to use the common title. CheeseDreams 23:39, 2 Dec 2004 (UTC)

[edit] Hausdorff dimension/Hausdorff measure

I think to split it.Tosha 06:09, 28 Dec 2004 (UTC)

Do uou mean you think you think are going to split the article? What are the advantages of doing so? Particularly, since the two subjects are tied so closely together.CSTAR 06:38, 28 Dec 2004 (UTC)

They are not tied, one is defined using the other Tosha 21:39, 28 Dec 2004 (UTC)

They are not tied
That's certainly a new viewpoint.
I'm certainly aware that one is defined using the other. But maybe I should have asked how you propose to split them up, particularly, as I'm sure you agree, logical order may have no relation to expository order. Note that the introductory paragraphs refer to Hausdorff's idea of using measure to determine some kind of dimension as opposed to some more naive box counting dimension. CSTAR 22:10, 28 Dec 2004 (UTC)
P.S. I don't oppose splitting them up, it's just that I don't see the expository advantages. But I'm convincible. CSTAR 22:23, 28 Dec 2004 (UTC)

I think it is a good idea to split them. The simplest definition of Hausdorff dimension does not involve the Hausdorff measure, but involves the Hausdorff content. Anyway, the Hausdorff content should be discussed somewhere. I think there is a lot to say about Hausdorff measure and it would not be good to put all this stuff here. Oded (talk) 19:52, 9 May 2008 (UTC)

Related material which should be discussed somewhere are doubling, Ahlfors regularity and Roger's generalizations of Hausdorff measure. Trying to find a rational way of organizing and subdividing all this stuff is difficult. If you want to have at it, go ahead.--CSTAR (talk) 20:37, 9 May 2008 (UTC)
I'm far from an expert about H-measure. I will probably start out by splitting H-measure away from here, put here the definition of H-dim in terms of H-content, create a page with the basic definition of H-measure. But then I would leave it to people more knowledgeble to write the rest. Oded (talk) 21:20, 9 May 2008 (UTC)

[edit] Does this really work? or?

Hausdorff dimensions, from left to right: ln(4) / ln(2) = 2.0, ln ( 4 ) / ln ( 2 ) = 2.0, ln ( 4 ) / ln ( sqrt ( 2 ) ) = 4.0.
Hausdorff dimensions, from left to right: ln(4) / ln(2) = 2.0, ln ( 4 ) / ln ( 2 ) = 2.0, ln ( 4 ) / ln ( sqrt ( 2 ) ) = 4.0.

I think this does not work correctly! Take a look at the image at right, there you can see three examples of Hausdorff dimensions. The first is a 2D Cantor set, four main attractors and scale factor Failed to parse (Cannot write to or create math output directory): 1:2 , no rotation. This gives a dimension of 2.0 and the result is also a two dimensional, 45º tilted square. The second image are the same but here a rotation of 45º is also used. From the Hausdorff method point of view this is still 2.0 dimensions but it's far from a two dimensional surface we see. Now if one choose to use a scale factor of 1:sqrt(2) then we vill get 4.0 dimensions and a perfec two dimensional surface. Is the Hausdorff method of counting dimensions wrong? Or did I miss something? ;-) // Solkoll 12:35, 1 Mar 2005 (UTC)

When you say "I think this does not work correctly!" what "this" are you referring to? CSTAR 12:44, 1 Mar 2005 (UTC)
This = the method of counting dimensions. The rotaion changes the dimesion from what I can see, (but I'm no pro, jus a guy who likes to make fractals and thinks a lot about how to do this, scale ratios and such). This is not taken in count using the Hausdorff method. // S
Rotating is a Lipschitz map. Read the the theorem on Lipschitz invariance! CSTAR 15:58, 6 Mar 2005 (UTC)
Why not try this: ln(points) / ln( | cos(angle) * scale | + | sin(angle) * scale | ). For the three images above this makes 2.0, 1.333..., 2.0 dimensions. (More like the looks of the pictures than 2.0, 2.0, 4.0 I say =) See also: the metric of the taxicab geometry: |x| + |y|, is this the same?. // Solkoll 11:59, 6 Mar 2005 (UTC)
Drawn with Fractal Attraction
Drawn with Fractal Attraction

Well, you are computing the similarity dimension (for an iterated function system), not the Hausdorff dimension. There is a theorem that says you get the Hausdorff dimension when you do this, provided the Open Set Condition (OSC) holds. Basically, this means there is not too much overlap of the images. In the first case, the OSC is true. You get a solid square, and it does have dimension 2. In the other two cases, the OSC is not true. In the second case, because of the overlap, the Hausdorff dimension could be less than 2. From the picture, it looks like it is. In the third case, the image is a solid square, but there is overlap again (can you see it in the picture?)... so the Hausdorff dimension (2) could be less than the similarity dimension (4).

--G A Edgar 15:19, 17 Mar 2005 (UTC)

Thank you for that professor Edgar =) this clears things out. And yes the overlaps is clearly shown in your picture. // Solkoll 16:42, 17 Mar 2005 (UTC)

[edit] New picture

I don't want to get into an discussion of the merits of pictures, but I don't particularly like the new SIrepinski image.CSTAR 12:49, 1 Mar 2005 (UTC)

Too kitschy I guess? too many colours? =) The old image was not good either, too pixly for my taste. I can make a softer Hi-res anti-anilise = no pixels - just lines, if you like? Mono-colour? B&W? RGB-fade as the current image but a clear backgrund? Make a wish =) // Solkoll 20:56, 1 Mar 2005 (UTC)
Two colour, uniform background. (BTW I made the previous one with XFIG). CSTAR 22:52, 1 Mar 2005 (UTC)
Better now? I'm using C/C++ under win32 & DirectX to create my images. This means a lot of work but also full control over the situation. If you, as I do, like kitschy fractal images? then take a look at commons:Category:Images of fractals or sv:Kategori:Bilder av fraktaler, (same sentens but in Swedish). All images are from my tools, (and all of them are not kitchy either =) // Solkoll 11:22, 2 Mar 2005 (UTC)

[edit] Additivity

"The Hausdorff outer measure Hs is defined for all subsets of X. However, we can in general assert additivity properties, that is

   H^s(A \cup B) = H^s(A) + H^s(B)

for disjoint A, B, only when A and B are both Borel sets. From the perspective of assigning measure and dimension to sets with unusual metric properties such as fractals, however, this is not a restriction."

This is wrong, it is for measurable A, B, not only Borel. The set of all measurable sets is strictly larger than the set of all Borel sets. This is true for at least Hausdorff and Lebesgue measure (proof can be by construction using inverse images of the Cantor ternary function, or by cardinality).220.245.178.131 05:37, 28 February 2006 (UTC)

It is certainly true for Borel sets. The intent of the claim is that some additional restriction is necessary on the sets A, B. I will fix the statement so it is fully accurate.
BTW one cannot conclude that in general if H^s(A \cup B) = H^s(A) + H^s(B) THEN both A, B are measurable. In fact, a set is E measurable in the Caratheodory sense iff it splits every set Ainto two parts so that the outer measures add up, that is for all A
\mu^*(A) = \mu^*(A \cap E) + \mu^*(A \setminus E)
--CSTAR 05:58, 28 February 2006 (UTC)

[edit] Recent edits

Why are these recent edits an improvement? Any subset of a metric space is a metric space. Moreover, it's thge behavior of the measure which is unintuitive, not the values. I think the article should be reverted to version of August 26 [1].--CSTAR 14:44, 12 October 2006 (UTC)

After rereading my changes I agree. I am reverting as you suggested. —Tobias Bergemann 18:03, 12 October 2006 (UTC)

[edit] finite hausdorff dimension for compact sets?

"For instance R has dimension 1 and its 1-dimensional Hausdorff measure is infinite." The first reason I would imagine that R is able to have infinite Hausdorff measure is that it is not compact. Is there a compact set whose Hausdorff measure is infinite, or a proof that such a set cannot exist? It would be a nice little comment to add after the quoted sentence either way.69.215.17.209 16:21, 31 March 2007 (UTC)

[edit] Hey

Hausdorff dimension eh, has anyone got a fractal with a hausdorff dimension = golden ratio? 83.70.46.38 06:21, 23 April 2007 (UTC)

[edit] About Hausdorff dimension and Hausdorff measure

The two concepts are related but are not the same: Hausdorff measure of smooth sets coincides with Lebesgue measure, while the Hausdorff dimension coincides with the "algebraic" dimension (very rougly, it is the cardinality of the set of linearly independent unit vectors needed to span the smallest linear subspace enclosing the given set). The confusion derives from the fact that in classical references, Hausdorff dimension is also called Hausdorff dimension measure: I have reported two links from the online Springer Enciclopedya of Mathematics clarifyng the two conceps.

I also remember that the fundamental reference in the field, i.e. Herb Federer's masterpiece, clearly states the two concepts and their difference (I'm quite sure and I will check later). In my opinion it is necessary to give rise to two separate voices. Daniele.tampieri (talk) 14:43, 7 April 2008 (UTC)

Hausdorff measure is a (a family of) countably additive measures Hλ on Borel sets, indexed by a real parameter λ . Hausdorff dimension is a real number defined in terms of Hausdorff dimension(measure Note Correction --CSTAR (talk) 03:17, 28 April 2008 (UTC)). All Hausdorff measures on Rn defined in terms of continuous metrics on Rn which are invariant under translation, are identical up to a nonzero scaling constant. In some instances, determining the value of this constant can be very important and Federer does a very careful job of computing these values. Determining the scaling constants is tantamount to computing volumes of spheres, and for some problems in mathematics the asymptotic behavior of these volumes as n goes to infinity is very interesting and important. However, for determining the Hausdorff dimension of sets, the value of the scaling constant is of no interest.--CSTAR (talk) 15:42, 7 April 2008 (UTC)
This is wrong, e.g. the metric d(x,y)^s for d Euclidean distance 0<s<1 is invariant under translation and continuous but the Hausdorff measures aren't the same. Oh well.--CSTAR (talk) 20:57, 5 May 2008 (UTC)

[edit] Date of introduction

As far as I can see the earliest reference was published in 1919. The fact that Hausdorff submitted the paper in 1918 doesn't mean the theory was "introduced" then, as the introduction currently states. The current text could be retained if Hausdorff was presenting his work at conferences in 1918 before publication of the article. —DIV (128.250.80.15 (talk) 02:44, 28 April 2008 (UTC))