Talk:Hausdorff dimension

Talk 2003 -2006

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"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

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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).

Banner

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

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)

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

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|ηυωρ]] 20:39, 26 Nov 2004 (UTC)

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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|ηυωρ]] 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)

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)

How is it not a measurement? What measure is not just a number assigned to a set?

JumpDiscont (talk) 02:50, 3 October 2009 (UTC)

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 Measure 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|ηυωρ]] 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)

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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|ηυωρ]] 20:33, 26 Nov 2004 (UTC)

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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)

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)

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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)

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.

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 $1:2$, 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

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Rotating is a Lipschitz map. Read the the theorem on Lipschitz invariance! CSTAR 15:58, 6 Mar 2005 (UTC)

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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)

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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)

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)

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)

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)

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)

Sure. Just take your favorite countable, compact metric space (e.g. 1/n with 0). Being countable, it has Hausdorff dimension 0, but the d=0 Hausdorff measure is just the counting measure, so it has infinite measure. —Preceding unsigned comment added by 72.33.108.253 (talk) 17:51, 26 November 2008 (UTC)

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)

Yes: Beardon (1965, On the Hausdorff dimension of general Cantor sets) showed that generalised Cantor sets with arbitrary dimension can be generated. Colin Rowat (talk) 07:07, 24 October 2011 (UTC)

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.

• Hausdorff measure
• Hausdorff dimension

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 H^λ 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)

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))

Informal discussion

Whoa. The "Informal Discussion" is anything but. Someone needs to bring this down to a level where someone who isn't a math major can understand. Or perhaps consider changing the name. Timothyjwood (talk) 04:37, 7 July 2008 (UTC)

I think Timothy is right. This informal discussion discusses too many things (e.g. topological dimension) that are not essential to the main point. We should rewrite the informal discussion to motivate the definition of Hausdorff dimension (going through Minkowski dimension). I envision something similar to an expanded version of what is currently the third paragraph. Oded (talk) 05:43, 7 July 2008 (UTC)

It's three years later, and the informal discussion is not clear to me: "But you can easily take a single real number, one parameter, and split its digits to make two real numbers." What does "split lll digits" mean? What is the grammar here: Is "one parameter" an appositive of "single real number"? If not, then should it read "single real number and one parameter and split"? I don't know. 211.225.34.164 (talk) 10:16, 28 June 2011 (UTC)

Another ~4 yrs, and still no progress in the lede and mentioned informal section. Perhaps in maths this is the equivalent of a secret handshake, that we simply are not intended to understand or enter this rarified realm. 165.20.108.150 (talk) 23:34, 5 March 2015 (UTC)

I can explain what "split its digits" means: take a number like 3.14159..., and split it into two numbers: one whose digits are those corresponding to even powers of 10 in the original number (3.45...) and one whose digits are those corresponding to odd powers of 10 in the original number (0.119...). Obviously, if you start with two different numbers and split them up this way (assuming you're careful about picking your decimal expansions) you'll get different pairs of numbers at the end; and any pair of numbers can be generated this way (since you can just take the two decimal expansions and interleave them). I don't know how to explain it more simply than just giving an example, though. 68.113.159.26 (talk) 04:08, 7 October 2016 (UTC)

Minkowski dimension, rather than Hausdorff?

[In the Section entitled "Informal discussion," i]s the construction given in the paragraph beginning, "Consider the number N(r) of balls of radius at most r ..." not that used in deriving Minkowski dimension, rather than Hausdorff? The key detail is the use of regular elements, the r-balls. If so, this gives an upper bound for Hausdorff dimension, which is typically much more easily derived than the Hausdorff dimension itself. 147.188.129.99 (talk) 20:30, 27 October 2011 (UTC)

Product inequality

Anyone have a proof (or sketch) that the dimension of a product is at least the sum of the dimensions?

Formal definition is wrong

I'm not an expert in this area, but I'm pretty sure the given definition of Hausdorff content is incorrect. It disagrees with the books I've looked at, and moreover does not make sense. The given definition places no restriction on the balls used to cover the set. The typical definition of Hausdorff content is to say something like: H^d_delta is the infimum of sums of (r_i)^d, where the set can be covered by balls B_i of radius r_i LESS THAN DELTA. Then define H^d of the set to be the limit of H^d_delta as delta goes to 0. Otherwise, for instance you get that the 1-dimensional Hausdorff content of the unit ball in R^3 is finite. Kier07 (talk) 16:36, 6 February 2009 (UTC)

Never mind, sorry. Though the words "content" and "measure" are very similar, so I think the article should be clearer about the fact that these are different ideas. Kier07 (talk) 16:44, 6 February 2009 (UTC)

Hausdorff content (defined here without delta) and Hausdorff measure (defined in many sources with delta) are, indeed, different. For example, the unit ball in Euclidean space R^n is covered by some ball of some radius, so the d-dimensional content is finite for all d. But of course the d-dimensional Hausdorff measure is finite only if d >= n.G A Edgar (talk) 11:21, 26 July 2009 (UTC)

Furthermore, defining Hausdorff dimension using the Hausdorff content is wrong. It must be defined from the Hausdorff measure.G A Edgar (talk) 11:21, 26 July 2009 (UTC)

OK, the definition of Hausdorff dimension is not wrong. Although the Hausdorff content and the Hausdorff measure are different, when one is zero so is the other. G A Edgar (talk) 13:34, 30 July 2009 (UTC)

Formal Definition

From the definitions at the beginning at the paragraph, it seems clear that we are interested in the Hausforff dimension of S. However, the definition at the end is given with X, which is the enclosing metric space and not what we were interested in in the first place. Wouldn't it be clearer and more correct to give it with S instead? Joelthelion (talk) 10:22, 17 August 2010 (UTC)

"Every space filling curve hits some points multiple times"

This assertion appears in the article. I can't tell if it's wrong or just confusing. None of the space-filling curves listed on the Space-filling curves page intersect themselves and the ones pictured have a Hausdorff dimension of 2.

I suggest removing or clarifying this assertion, but I don't have the mathematical expertise to know which is appropriate.

-

That statement is correct and here's why: A space filling curve maps a line to a filled square or cube or hypercube. The definition gives us that the curve is both continuous and onto. If it were to not intersect itself, then the curve would also be 1-1, which in turn become an isomorphism. Since a line and a square are not isomorphic, the curve should hit some points multiple times.

It seems counter-intuitive; but it's correct nonetheless. 212.175.32.133 (talk) 09:26, 7 April 2015 (UTC)

Article tag added

Yet another mathematical encyclopedia completely inaccessible to a lay reader? Tag added because not even the lede is approachable. This is a typical maths article problem, that mathematicians are writing for themselves. I challenge contributing editors to write a lede and figure legends (if not the whole of the introductory portions of the article) that any high school-educated reader can read—without chasing every wikilink that appears. Wikilinks and citations are intended to allow interested readers to go deeper in particular subject areas, not to replace the need for good, understandable exposition. 165.20.108.150 (talk) 23:20, 5 March 2015 (UTC)

Lede rewritten from other Wikipedia article content and new sources, making at least opening more readable. See how this fits. 71.239.87.100 (talk) 04:57, 6 March 2015 (UTC)

Tagger let me suggest reading the whole Wikipedia article on Mathematics. If after doing that you still come back expecting mathematicians to write their Wikipedia articles on the subject in such a way that anyone with a high school diploma can instantly understand them, then sorry, I can't help you. By the way, if this particular subject didn't happen to have the loaded term "dimension" in its name, you likely would have never added the tag. (Would you have added it to any page with a title like "Diophantine equation" or "Kuratowski closure-complement problem" instead? I doubt it.) Tag removed. Mathematrucker (talk) 13:06, 13 July 2017 (UTC)

References

These are the sources that appear for the Hausdorff dimension at its entry at the Encyclopedia of Mathematics, see [ http://www.encyclopediaofmath.org/index.php/Hausdorff_dimension], accessed 5 March 2014:

• L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
• K.J. Falconer, "The geometry of fractal sets" , Cambridge Univ. Press (1985) MR0867284 Zbl 0587.28004
• H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
• F. Hausdorff, "Dimension and äusseres Mass" Math. Ann. , 79 (1918) pp. 157–179 MR1511917
• W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)
• P. Mattila, "Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005

71.239.87.100 (talk) 01:33, 6 March 2015 (UTC)

Citations inappropriate moved here for discussion

Given the WP:PSTS policy strictures on use of primary sources, the following two bits of text and their sources are moved here for discussion. In my view they add nothing to the article, and begin a pattern of dumping of trivial results here that will in long run, move the article to a status even further from GA than it currently is:

• The bond system of an amorphous solid changes its Hausdorff dimension from Euclidean 3 below glass transition temperature T_g (where the amorphous material is solid), to fractal 2.55±0.05 above T_g, where the amorphous material is liquid.

REF: M.I. Ojovan, W.E. Lee. (2006). "Topologically disordered systems at the glass transition". J. Phys.: Condensed Matter. 18 (50): 11507–20. Bibcode:2006JPCM...1811507O. doi:10.1088/0953-8984/18/50/007.

[objection: very narrow primary research result, of questionable general importance, not further supported by a secondary source, and not integrated into any clear theme of the overall article (i.e., a stray related research result that lacks notability for inclusion)]

• SECTION: The Hausdorff Dimension Theorem. For any given r ≥ 0, and integer n ≥ r, there are at least continuum-many fractals $A\subset \mathbb {R} ^{n}$ of Hausdorff dimension r.

REF: Soltanifar, M. (2006). On A Sequence of Cantor Fractals, Rose Hulman Undergraduate Mathematics Journal, Vol 7, No 1, paper 9.

[objection: very narrow primary research result, of questionable general importance, not further supported by a secondary source, and not integrated into any clear theme of the overall article (i.e., a stray related research result that lacks notability for inclusion); moreover, a publication of primary results in an undergraduate university journal, and not a standard, high quality venue, even for primary results, so clearly in need of substantiation before reintroduction]

Le Prof 71.239.87.100 (talk) 04:56, 6 March 2015 (UTC)

Examples section

Why is the examples section flooded with "citation needed"s? These examples aren't some hard problems which took hundreds of years to solve, at maximum they would take an average person ten minutes to verify with a pen and paper. 212.175.32.133 (talk) 11:28, 6 April 2015 (UTC)

I reacted in the same way to-day (more than 18 months later, with nothing really having happened since that time). Here are the edits where they were introduced. @Leprof 7272: can you confirm that they are your edits?

The edits not only cluttered this and other sections with "unsourced" tags (for both sections and sentences), it also made some re-labeling at the references. Before these edits, the reflist appeared under the heading Notes, and references more comprehensive literature under the heading References. This was changed to References and Further reading, respectively.

My personal opinion is that the article was better before than after these edits. However, Leprof 7272, or whoever else the editor was, was at least formally following some WP standard. The question more is if this really is a good standard, or a good interpretation of the standards. At least for math articles, I think that more general references for whole sections are better than sourcing each and every fact separately - and especially so, when a subject is treated, where indeed there are numerous interesting but fairly immediate consequences of the definitions, as in this case. If you have understood the definitions, then you also should understand the consequences, which now are individually tagged as unsourced, with reasonably little trouble. (If you never in your life have met similar subjects, and still on a theoretical level understand the definitions, then it may take a little longer than 10 minutes to verify them; but with a general reference provided, you also just might try to look them up.)

Actually, I have the impression that WP editors with more mathematical experience in general tend to be a bit more flexible about referencing each and every obvious consequence of the definitions separately. However, it is possible that this flexibility deviates from WP standards, at least as these are interpreted by a non-ignorable fraction of the experienced editors with mostly non-mathematical main interests.

In this case, I'd like to remove a number of the tags; but not if there is a strong dissent. JoergenB (talk) 20:12, 27 October 2016 (UTC)

As I mentioned above in my justification for removing the inappropriate "too technical" tag (how it remained for over two years is a mystery to me), I suspect that all the tag clutter here may be partly attributable to the cachet the word "dimension" has in ordinary English. Mathematrucker (talk) 13:41, 13 July 2017 (UTC)

Important omission

The famous formula $D={\frac {log(P)}{-log({\frac {1}{\epsilon }})}}$ is absent!

If the article should be reduced to one formula, it would be that formula, and is not clearly and explicitly exposed. — Preceding unsigned comment added by 206.132.109.103 (talk) 14:07, 5 January 2017 (UTC)

"Balls" misleading?

In the "Formal Definition" section, I'm assuming that the "balls" are balls under the supremum measure? I don't think that would be at all obvious to the average reader... Twin Bird (talk) 03:26, 5 April 2020 (UTC)

Adding a Generalized Hausdorff Dimension Theorem to the Page

Hi Editors; I suggest adding this Theorem to the page for readers' reference. It shows the existence of deterministic fractals for given Hausdorff dimension and Lebesgue measure. Thanks\\

P.S.\\

Theorem 6 (The Generalized Hausdorff Dimension Theorem). For any real r>0 and l≥0, there are aleph-two (symmetric) fractals with the Hausdorff dimension r.1{0}(l)+n.1(0,∞)(l) and Lebesgue measure l in Rn where (⌈r⌉≤n).

Link:

https://www.mdpi.com/2227-7390/9/13/1546 https://doi.org/10.3390/math9131546 — Preceding unsigned comment added by MohsenSoltanifarNA (talk • contribs) 02:21, 9 July 2021 (UTC)

British Coast Line

The approximation of the British Coast Line looks like an approximation of the Minkowski dimension (that often coincides with the Hausdorff dimension). For Hausdorff dimension balls of different size < delta are allowed. Maybe that should be clarified. — Preceding unsigned comment added by 2A02:908:621:3520:FFDE:36A8:F977:BA99 (talk) 18:47, 16 April 2022 (UTC)