Talk:Fractal dimension

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Introduction

The fractal dimension of an object is clearly not always, and never is, greater than the dimension of the space containing it. By definition the dimension of an object is less than the dimension of the space containing it.

Analogy to Lines and Planes

The attempted analogy to lines and planes is utter nonsense; the length of the Koch curve is the same as any line: infinite, and the dimension in the conventional sense (coordinates ~ degrees of freedom) is precisely 1. Fractal dimension is an entirely different concept. Augur (talk) 06:29, 10 January 2011 (UTC)[reply]

Actually I've seen before attempts to intuitively explain fractal dimension describing a fractal line as an object somewhat between a line and a plane, so that paragraph is not a new concept; I'm sure there should be possible to find some reference covering that intuition. If you read again the bit about the infinite line, it doesn't refer to the line as a whole but to the segment of the curve between two points, which for non-fractal curves is always finite.Diego Moya (talk) 11:36, 10 January 2011 (UTC)[reply]

This has been clarified.Akarpe (talk) 13:25, 1 February 2012 (UTC)[reply]

I've reverted and reworked some of your changes to the introduction section as I find them too abstract. A concrete example is usually beneficial for understanding so I have reintroduced it. Diego (talk) 10:56, 4 February 2012 (UTC)[reply]

yeah; I actually had moved it and condensed it but I like it better where you put it backAkarpe (talk) 12:58, 5 February 2012 (UTC)[reply]

Expand and make simpler please

Expand and make simpler please Alan2here 23:06, 5 November 2006 (UTC) I agree. Please define terms such as "attractor" for non-mathematicians. Also in figure 1 should r be l? —Preceding unsigned comment added by 71.164.135.92 (talk) 05:16, 20 December 2007 (UTC)[reply]

That's a tricky request, the picture is a little vague as stands, I'll try to clean that up. The attractor is the limit of the fractal, ie. the black triangles in the Seprinski gasket if we could iterate infinitely. I really don't think the major content of this article would be useful to describe to non-mathematicians, but the introductory paragraphs could do with a simplified layperson summary. I'll see what I can do. Nazlfrag (talk) 04:20, 31 December 2007 (UTC)[reply]

I've altered the first picture slightly, replacing r= with l= as per the main body of text and adding the values that equate to 1. As for a better description of attractors, I left that to the attractor article and linked the first occurrence to said article. As for a summary, the description in the second paragraph of the Koch curve is succinct, accurate and a better description than any I can think of. Perhaps an example with a real world fractal would help the non-mathematicians, though the concept of fractional dimensions is hard enough for mathematicians to get their heads around in the first place. Where is Mandelbrots amazing descriptive ability when you need it! The Fractal Geometry of Nature is comprehensible to nearly everyone, we should strive for that same level of clarity. Nazlfrag (talk) 05:52, 31 December 2007 (UTC)[reply]

Well, there are no (known) true fractals in nature. Fractal models for e.g. the coast of Norway may work fairly nicely for a while, but when you get down to subatomic levels, they're out. You run into trouble with quantum theory, I'm afraid. Similar obstacles hold for (the original) Brownian motion; and e.g. stock market behaviour suffers from another kind of "quantisation": You have no "movement" at all in the fractions of time where there is not a single transaction finished. Nevertheless, perhaps these three classical examples might serve as illustrations of the kind you're asking for (with an appropriate warning against overemploying the fracta models).

The second paragraph Koch curve unhappily uses "curve" in two different meanings, yielding the slightly paradoxical statement that the curve is not a curve. Apart from this minor matter, I agree with Nazlfrag's approval of that paragraph. JoergenB (talk) 15:46, 1 January 2008 (UTC)[reply]

I agree that some clarification is needed in this and similar articles. For example, the expression D=(log N(l))/log l should have it's symbols, N, l, introduced beforehand so that the uninitiated understands their meaning and behaviour. It is unclear, for example, if N=l^D (preceeding paragraph) is a definition, property, or otherwize. 68.144.80.168 (talk) 20:39, 30 October 2008 (UTC)[reply]

I've cleaned this up.Akarpe (talk) 13:25, 1 February 2012 (UTC)[reply]

Various definitions

Examples

I thnk the article is not bad. One suggestion for improvement: there are many definitions alluded to, and some discussed. Is it possible to show how differently the various definitions would quantify the fractal character of a given object? If there is some well-known (to mathematicians, at least) object where you could say, "Now by definition (1) we get 1.23 and by definition (2) we obtain 1.27, but definition (3) actually goes to infinity", or some such. —DIV (128.250.80.15 (talk) 06:45, 26 April 2008 (UTC))[reply]

The specific definitions section is now linked to the relevant pages but it still could use that bit about the value of one fractal by all the methods.Akarpe (talk) 13:25, 1 February 2012 (UTC)[reply]

Hausdorff

Can the parenthetic comment in the article "(which is more or less the Hausdorff dimension)" be elaborated on just a little more, perhaps in a footnote? —DIV (128.250.80.15 (talk) 06:48, 26 April 2008 (UTC))[reply]

The parenthetic comment has been modified and the required footnote added. (87.4.47.185) — Preceding unsigned comment added by 87.4.47.185 (talk) 19:56, 1 October 2011 (UTC)[reply]

More general introduction?

I would think that the introduction could be more general, in that regular dimension are a 'subset' so to speak of fractal dimensions, since they can be represented when D is not a fraction or irrational number (i.e. and integer). This would nicely connect fractal dimensions to 'regular' dimensions, since they probably have something to do with one another, but most of the time they are treated as 'disjoint' areas of knowledge. Rhetth (talk) 21:54, 4 March 2009 (UTC)[reply]

Done!Akarpe (talk) 13:25, 1 February 2012 (UTC)[reply]

Unclear step =

It is not really clear how you get from the expression D=log N(l)/ log l to D=lim epsilon->inf log N(epsilon)/log 1/epsilon. Could you please explain this more in detail? Thank you. —Preceding unsigned comment added by 134.221.149.67 (talk) 08:36, 20 August 2010 (UTC)[reply]

I ran into that problem as well, and I think, looking at the later equations, that "log l" should be "log 1/l". Since no one has answered your question, I'm inclined to go ahead and fix it, and hope that if I'm wrong, someone who understands better will correct the problem.Huttarl (talk) 21:43, 20 October 2010 (UTC)[reply]

There seems to be a sign problem here. The expression, N=L^D, seems right. e.g. the square in fig 1 divided into 4 has N=4, L=2 and D=2 which fits OK (I'm writing L, not l which looks like 'one'). Taking logs gives logN=D.logL, whence D=logN/logL, not D=logN/log(1/L) as written in the article. With the same example from fig 1, D=log(4)/log(2)=+2 (right) whereas D=log(4)/log(1/2)=-2 (wrong sign). One could write D=-logN/log(1/L) instead but I prefer to leave amending the article to someone who knows about fractals.Ajrc (talk) 22:51, 7 November 2010 (UTC)[reply]

Fixed.Akarpe (talk) 13:25, 1 February 2012 (UTC)[reply]

Broken layout

Akarpe, your recent edits have created a broken layout and an introduction section that is too long. It also includes too many changes at once, making it impossible to fix them one by one. I've reverted the article to the previous version and saved your edits here so that it can be properly fixed, before placing them into the main article. Please refrain to include them again until we discuss them here. Diego (talk) 14:22, 6 February 2012 (UTC)[reply]

P.S. Please explain what changes are you introducing and we will agree on them one by one. The big images all at the article start are a problem since they're too big and too many. Also the I find the new introduction too long and comlex and prefer the simple introductory definition that was in place. What changes are you trying to introduce? Diego (talk) 14:25, 6 February 2012 (UTC)[reply]

Holy moly! That was crazy. I removed an image then it came back; I changed image size and it reverted! We were working simultaneously and it really messed things up. I was making it accessible, clarifying some essentials of the fractal dimension, making it useful to the layperson, especially concepts related to the ideas about space filling, providing illustrative images. I also added a history section. It got cooked, though, in the simultaneous exchange. I was trying to incorporate feedback from a few people who read it and subsequently misunderstood what they read; my goal was to provide missing information and correct misleading information. Alas...

Akarpe (talk) 14:48, 6 February 2012 (UTC)[reply]

I did did a small change to two images to remove their fixed size; that's not the problem. The big section at the top with 9 images at least 300px wide is. If your friends have problems seeing the images, they should use their browser's zoom function, not sqeezing the article text between several poster-size images. And for the feedback for those people, would you mind writing here what are they concerns? This way we can write a new intro section that is in accordance to the lead section guidelines. Diego (talk) 14:55, 6 February 2012 (UTC)[reply]

wow! The images were not showing up large; in fact, they were all very compact; I must have a different browser than you. That is good to know. Poster size is silly, definitely not what I saw on my screen. sorry! the accessbility part was for a friend who is fully blind; especially the explanation in the Koch curve, which is easier to grasp than the snowflake for many reasons, one being that it is a line but the flake resembles a circle so people get confused when trying to perceive of it as a line of dimension 1 - they intuitively think it is a surface but in this case it is a 1 dimensional object even if it does join to itself; thus, the line is not only more like what von Koch actually published in his 1904? paper, which is hand drawn but gets the point across.Akarpe (talk) 15:11, 6 February 2012 (UTC)[reply]

the changes to the introduction were to explain what the fractal dimension represents; to the laypeople who commented, they did not get what the previous version meant so we worked on it through multiple edits to ensure that they got their questions answered in the text. They said the dimension thing was hard to grasp without already knowing it; I'll do edits in shorter bits and keep the images to style manual default sizes and not sandwich - didn't realize it showed up differently elsewhereAkarpe (talk) 15:44, 6 February 2012 (UTC)[reply]

That's great. I also recommend that you keep the images at different sections of the article instead of having all them together at the top. This will ensure that they work well at all image sizes, browsers and mobile devices. As for the text, I think your version would work best as an Introduction section below the lead. The lead section should be kept short and cover the main article points in summary style. Any detail (like the one given after "To elaborate...") is better in the article body. Diego (talk) 16:00, 6 February 2012 (UTC)[reply]

I put the images in at the sections as recommended; thanks for the tip as I thought putting them at the top in a group stacked them neatly and made it easy to track figure numbers in case of edits, but I agree it is better in the pertinent section. I put the intro section in after the lead.Akarpe (talk) 06:38, 7 February 2012 (UTC)[reply]

too big to be a one-dimensional object, but too thin to be a two-dimensional object

I wonder if this gives a lot of readers the impression that fractal sets exist in a mysterious dimension, cuz it did to two people I asked to read the page. But fractal sets have 0 or 1 or 2 or 3 topological dimension just like the rest of the sets in geometry do. Having a fractional dimension doesn't change the set's topological dimension but the phrase seems to say it does. Would it be better reworded? — Preceding unsigned comment added by Akarpe (talk • contribs) 07:05, 8 February 2012 (UTC)[reply]

Invalid statement in lead about dimension?

The lead says "a fractal dimension is greater than the dimension of the space containing it". This seems wrong: it is greater than the object's topological dimension, and (AFAICT) never greater than the dimension of the space containing it. Comment? — Quondum ^☏ 19:09, 14 July 2012 (UTC)[reply]

Yes, I agree - that statement apeears to be wrong. I have removed it. Gandalf61 (talk) 12:46, 15 July 2012 (UTC)[reply]

Perhaps this picture can save you the proverbial kiloword

http://gosper.org/wikifrac.gif Bill Gosper (talk) 13:13, 2 November 2013 (UTC)[reply]

This image intrigues me quite a bit, since the dimension of the Koch curve seems to be $log_{3}4$ , not $log_{7}9$ . These seem to be fundamentally different ways for defining fractal dimension. Or perhaps there is just imprecision in the picture and I am right to be sceptical about it.Warichnich (talk) 09:39, 26 June 2017 (UTC)[reply]

Definitions of dimensions

There is no such thing as a synchronized limit of two variables in mathematics, there are actually some examples why this is not possible. This means that the definition of the correlation dimension is not well-defined. I changed it to a twice limit, the one that actually could make sense, but since I am not sure of this at all I added a citation needed. I was reluctant to remove it since the general idea is clear nevertheless, even with errors in the formula. Hope this is in line with all the Wikipedia policies I don't know too much about yet. The other thing is, I don't really see the point of putting in the formula of a definition without defining the elements in it (cf. Higuchi dimension) but perhaps ideally someone who is fit in such matters could do that. I don't understand the source well enough.Warichnich (talk) 09:39, 26 June 2017 (UTC)[reply]

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Relative time reference should be fixed.

In the "Estimating from real-world data" section, the statement, "Nonetheless, the field is rapidly growing..." is not anchored to any dates. Was this written 10 years ago? It likely will not be correct 15 years from now. And I'm just curious about the answer anyway. EvolutionOfTruth (talk) 03:01, 12 July 2020 (UTC)[reply]

Conceptual error in section heading

The section heading "D is not a unique descriptor" is wrong.

Here a few ways to correctly describe D.

D is a unique descriptor

D is a fractal descriptor

D is a property of a given fractal

D is not a complete specifier

D is one descriptor of a fractal, but is insufficient to define the fractal

Feel free to wordsmith but make the heading conceptually correct! Wcmead3 (talk) 16:55, 6 January 2022 (UTC)[reply]

What does it mean to be "mythologized" in the introduction?

"It has also been mythologized as a measure of the space-filling capacity of a pattern that tells how a fractal scales differently and in a fractal dimension, i.e. one that does not have to be an integer" What does this sentence mean? Is it incorrect to say that the fractal dimension is a measure of space-filling capacity? Why is it a myth? 2601:18A:C67F:D540:DE2:6CA4:8A5A:594D (talk) 01:57, 28 January 2023 (UTC)[reply]

I think this should be replaced with "it is also a measure of," because fractal dimension is, literally, a measure of how the object's volume changes as it scales. E.g., a line has fractal dimension 1 because when you scale it by 2x, the "volume"(length) increases by 2x. Scale a square by 2x, "volume"(area) increases by 4x, or 2^2x. Scale a cube by 2x, volume increases by 8x, or 2^3x. Then if you scale a 1.5d object by 2x, the "volume" (a 1.5 dimensional measure) would increase by ~2.82x, or 2^1.5x. This is what it says in the "Role of Scaling" section. Mrfoogles (talk) 03:25, 20 June 2023 (UTC)[reply]

I'm just going to edit it to say "it is also" (I don't understand fractal dimension super well but I do know the definition) Mrfoogles (talk) 03:26, 20 June 2023 (UTC)[reply]