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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
Knowledge 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.
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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.
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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.
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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...
557:; 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).
1358:"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?
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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
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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,
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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
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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
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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
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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
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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
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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
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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)
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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
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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.
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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
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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
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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
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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
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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,
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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
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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.
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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.
1374:"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.
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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
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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)
824:, 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
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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.
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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.
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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
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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 elsewhere
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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.
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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? —
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1139:. 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.
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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.
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Can the parenthetic comment in the article "(which is more or less the
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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.
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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.
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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
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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?
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since they probably have something to do with one another, but most of the time they are treated as 'disjoint' areas of knowledge.
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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)
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non-mathematicians, but the introductory paragraphs could do with a simplified layperson summary. I'll see what I can do.
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Now by definition (1) we get 1.23 and by definition (2) we obtain 1.27, but definition (3) actually goes to infinity
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doesn't change the set's topological dimension but the phrase seems to say it does. Would it be better reworded?
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It is not really clear how you get from the expression D=log N(l)/ log l to D=lim epsilon-: -->
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inf log N(epsilon)/log 1/epsilon. Could you please explain this more in detail? Thank you.
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yeah; I actually had moved it and condensed it but I like it better where you put it back
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This image intrigues me quite a bit, since the dimension of the Koch curve seems to be
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If you found an error with any archives or the URLs themselves, you can fix them with
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D is one descriptor of a fractal, but is insufficient to define the fractal
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The parenthetic comment has been modified and the required footnote added. (
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Yes, I agree - that statement apeears to be wrong. I have removed it.
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Feel free to wordsmith but make the heading conceptually correct!
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The section heading "D is not a unique descriptor" is wrong.
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What does it mean to be "mythologized" in the introduction?
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for additional information. I made the following changes:
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Perhaps this picture can save you the proverbial kiloword
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137:High-priority
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62:High‑priority
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986:— Preceding
981:
892:guidelines.
890:lead section
864:
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688:— Preceding
681:
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386:Introduction
364:
324:
264:Chaos theory
241:
201:
136:
96:
40:WikiProjects
1055:Bill Gosper
751:—Preceding
694:87.4.47.185
684:87.4.47.185
486:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
1409:Categories
1284:Report bug
1161:Warichnich
1141:Warichnich
418:Diego Moya
1390:Mrfoogles
1376:Mrfoogles
1267:this tool
1260:this tool
1033:Gandalf61
1017:dimension
663:Hausdorff
482:Alan2here
1273:Cheers.—
1000:contribs
988:unsigned
771:problem.
753:unsigned
690:unsigned
622:Examples
566:JoergenB
562:Nazlfrag
528:Nazlfrag
509:Nazlfrag
488:unsigned
1341:Wcmead3
1187:my edit
1021:Quondum
773:Huttarl
367:on the
340:Physics
331:Physics
287:Physics
244:on the
217:Systems
208:systems
164:Systems
139:on the
30:B-class
1103:, not
992:Akarpe
962:Akarpe
926:Akarpe
911:Akarpe
883:I did
866:Akarpe
822:Akarpe
803:Akarpe
801:Fixed.
730:Akarpe
715:Rhetth
649:Akarpe
603:Akarpe
460:Akarpe
432:Akarpe
36:scale.
960:lead.
945:Diego
894:Diego
846:Diego
830:Diego
728:Done!
446:Diego
402:Augur
1394:talk
1380:talk
1364:talk
1345:talk
1305:talk
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788:Ajrc
777:talk
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607:talk
591:talk
570:talk
550:true
532:talk
513:talk
496:talk
464:talk
450:talk
436:talk
422:talk
406:talk
236:High
210:and
131:High
1241:RfC
1211:to
1201:to
686:)
359:Low
1411::
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42::
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