Talk:Hausdorff dimension

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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)
replies: text: Done. -- Jheald 22:01, 19 January 2006 (UTC)
thread_title:  Talk 2003 -2006 

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

message: 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)
thread_title:  Doubling property 

message: 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)
replies: text: 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)
replies: text:  Two colour, uniform background. (BTW I made the previous one with XFIG). CSTAR 22:52, 1 Mar 2005 (UTC)
replies: text:  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)
thread_title:  New picture 

message: 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 .--CSTAR 14:44, 12 October 2006 (UTC)
replies: text: After rereading my changes I agree. I am reverting as you suggested. —Tobias Bergemann 18:03, 12 October 2006 (UTC)
thread_title:  Recent edits 

message: "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)
replies: text: 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)
thread_title:  finite hausdorff dimension for compact sets? 

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

message: 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)
replies: text: 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)
replies: text:  This is wrong, e.g. the metric d(x,y)^s for d Euclidean distance 0= n.G A Edgar (talk) 11:21, 26 July 2009 (UTC)
thread_title:  About Hausdorff dimension and Hausdorff measure 

message: 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)
thread_title:  Formal Definition 

message: 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.
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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)
thread_title:  "Every space filling curve hits some points multiple times" 

message: 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)
replies: text: 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)

text: 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)
thread_title: Article tag added

message: 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)
thread_title: References

message: 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 Tg (where the amorphous material is solid), to fractal 2.55±0.05 above Tg, where the amorphous material is liquid.
: REF: 
:[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)
thread_title: Citations inappropriate moved here for discussion

message: 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)
thread_title:  Important omission 

message: 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)
thread_title:  "Balls" misleading? 

message: 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)
thread_title:  Adding a Generalized Hausdorff Dimension Theorem to the Page 

message: 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)
thread_title:  British Coast Line