Talk:Fractal/Archive 3

image
I have an image of a real-life fractal. It's from the glass door on a wood-burning stove. The charring on the glass flakes off in a fractal pattern. It's not a great picture, but it is a great example of fractal behavior in a physical process. Does anyone think it would help to contribute it to the article (and to the Commons)? --Cheeser1 07:51, 1 July 2007 (UTC)

Fractal-generating software
In the history of this article, a comprehensive list of fractal-generating software is buried. Based on this list, a dedicated article for the list of such software could be created.

See the revision of the article, after which the list of fractral-generating software was removed by the courageous User:Salix alba.

Above, an objection has been raised that "fractal-generating software" is a poorly defined category. I cannot confirm such a statement. To begin with, any program that generates a Mandelbrot set, Julia set, of IFS is a fractal-generating software. Put differently, to the extent to which the term "fractal" is defined, the term "fractal-generating software" is defined too.

--Dan Polansky (talk) 08:11, 23 February 2008 (UTC)

I personally believe that if you really want these links you can probably put them on your user page and reference this fractal page, otherwise I would suggest compiling a list somewhere outside wiki.

Wyvern917 (talk) 14:03, 20 August 2008 (UTC)

CD picture of "fractal" referenced to number 7
didn't find any mention in the source that microwaved CD's or DVD's exhibit fractal features. did you find any mention of that? --Kirils (talk) 12:14, 25 September 2008 (UTC)

Fractal software mention
I believe it is necessary to mention fractal generators. A person who knows little about fractals will be none the wiser after reading the four methods of making fractals. What a newcomer most likely wants to know is how fractals are made in practise, rather than in theory.

Further, I think that a short article on fractal generators is appropriate, and I have begun work on this.

If there are good reasons why this is unnecessary or inappropriate then please add them here, rather than just deleting. Soler97 (talk) 02:05, 4 October 2008 (UTC)
 * I do still object to its inclusion. Mostly on the grounds that it is so obvious of an observation as to be meaningless.  No one writes their own fractal programs or uses the mathematics to draw their own fractals.  Of course everyone uses a fractal generator.  This section merely presents the mathematics that those fractal generators use.  If it isn't made clear earlier that most people use generators then it could be, but the sentence does not belong in a section discussing the algorithms, in my opinion.TheRingess (talk) 22:02, 6 October 2008 (UTC)

Do you object if I put the same sentence at the end of the introductory section? In my experience, the obvious should be pointed out, especially to people who have no knowldege of fractals and very little of computers. It may be obvious to you and me, but people routinely ask me "How are fractals made?" These people are satisfied with the answer, "using a program". An encyclopedia article should presume the absolute minimum of knowledge on the topic in question.

Also, do you object to the addition of a short article on fractal generators? Soler97 (talk) 22:40, 7 October 2008 (UTC)

"Colloquial usage"
What exactly does the lead sentence mean by "In colloquial usage, a fractal is..."? How is "fractal" a colloquialism? See also colloquial. If the definition is wrong, fix the definition, don't label it a colloquialism. If the definition is correct (albeit rough or vague), then it's fine as-is (minus "colloquial"). --Cheeser1 06:15, 3 October 2007 (UTC)


 * The point that "colloquial" is trying to make is that what follows is not a mathematically precise definition of a fractal, because it is too vague. "Colloquial" is a flag to say "please don't pick holes in this definition, we know it is not exact". Unfortunately, we cannot put in a mathematically precise definition because there isn't one (AFAIK). Every attempt that I have seen at a precise definition either excludes objects that are generally agreed to be fractals, or includes objects that are generally agreed not to be fractals. But maybe "colloquial" is not the right term here - would "informal" be better ? Gandalf61 08:51, 3 October 2007 (UTC)


 * There isn't one precise definition of "integral" either. A better example, take "chaos." How about this:
 * A fractal is generally "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured".
 * Does that sound better? I'd also probably want to change "A fractal as a geometric object generally has the following features" to simply "A fractal often has the following features." You think that works? --Cheeser1 14:13, 3 October 2007 (UTC)


 * So long as it remains clear that the term has no really precise mathematical definition (Well, in fact it does: a fractal set has non-integer dimension!) But the point is that it has a more widely understood vernacular meaning (which is arguably much more important)--C G Strauss 21:20, 6 October 2007 (UTC)


 * Unfortunately, the "non-integer Hausdorff dimension" definition does not work, because it excludes objects such as the Smith-Volterra-Cantor set (dimension=1) and Peano curves (dimension=2), which are generally considered to be fractals. See List of fractals by Hausdorff dimension for more examples of fractals with integer Hausdorff dimension. Gandalf61 08:08, 7 October 2007 (UTC)


 * I agree - I think the imprecise definition is actually quite appropriate, given the different uses of the term to describe what are similar (if not precisely the same) phenomena. --Cheeser1 17:07, 9 October 2007 (UTC)

For a formal definition of a fractal, what about a set whose Hausdorff dimension differs from it's topological (Lebesgue covering) dimension? —Preceding unsigned comment added by 82.31.209.115 (talk) 22:21, 2 July 2008 (UTC)


 * It's one of the properties that a fractal can have. I don't know if there are counter-examples. Actually, the page states that "It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve)". I think that the text between parentheses is mathematically wrong, since the Hilbert curve as a topological dimension of 1 (being a curve) and an Hausdorff dimension of 2. Maybe it remains from an earlier version, where it was only stated that a fractal has a non-integer Hausdorff dimension? -- Clément Pillias (talk) 09:18, 4 November 2008 (UTC)

Fractal program on NOVA
NOVA recently aired a television program on fractals. (the full video of which is available to be viewed online, as well as other things pertaining to fractals) Perhaps the link should be mentioned in the External links section? http://www.pbs.org/wgbh/nova/fractals/ Cardsplayer4life (talk) 09:45, 13 November 2008 (UTC) The Nova presentation "Mysteriously beautiful fractals are shaking up the world of mathematics and deepening our understanding of nature" adds an additional dimension to the discussion. It brings to life many of the individuals who pioneered this form of geometry as well as brings to life practical application of the mathematics. I disagree that Wikipedia should limit the "Amount of Links" on a issue. The Nova presentation can also be embedded.

       Watch the full episode. See more NOVA.

More Examples from Nature
I wonder if anyone thinks it worthy of mention in the "In Nature" list of examples that gecko toes and feathers are fractal in structure? Pammalamma (talk) 23:02, 29 November 2008 (UTC)
 * Certainly you could mention these examples if you have a reliable source for both of them. Gandalf61 (talk) 17:21, 30 November 2008 (UTC)


 * Picture collection of natural fractals

Hilbert Curve
It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve). As far as I know Hilbert Curve has Haussdorf dimension=2 and topological dimension=1. Is that correct ? Lbertolotti 8 febb


 * No, the Hilbert curve (i.e. the image of the Hilbert map in R2) has a topological dimension of 2, equal to its Hausdorff dimension. I checked this at the Mathematics Reference Desk. Gandalf61 (talk) 18:00, 15 March 2009 (UTC)

nonfractals?
by the definition given in this article, the last five images, interesting as they are, are not fractals. why are they in this article?· Lygophile   has   spoken  15:12, 14 May 2009 (UTC)


 * Perhaps you could expand on that a little. Which specific images do you mean ? Why exactly do you think they are not fractals ? Gandalf61 (talk) 15:24, 14 May 2009 (UTC)
 * well, the very last five images in the article. the "poenix set" seems to merely contain a bunch of spirals (which are inherently perferct fractals), so i don't think that's really notable as a fractal image (but you don't get to see much of it). the "Pascal generated fractal" and the "fractal flame" are a in essence bunch of lines. if you zoom in, the lines get thicker and less curly, and it doesn't resemble itself in different zoom-scales. the Sterling fractal uses overlapping copies of an image, like a bunch of fractals do, but in contrast with those fractals, these images are not ordered in a way, that they collectively resemble that image, nor does that image have parts that resemble that image. it's basically the same for the julian one.· Lygophile   has   spoken  15:53, 14 May 2009 (UTC)


 * With regard to "zooming in", those images are just screen-shots of images created by fractal generating software programs. When you "zoom in" on the image, you are just expanding pixels - you cannot "zoom in" to show more detail. You seem to be expecting to "zoom in" in the image as if you were using the fractal generating program itself - that is just not possible. With regard to having "parts that resemble that image", I think you are talking about the property of self-similarity. That property can be quite difficult to determine from a single image - just because an image does not have any visually obvious self-similarities does not mean that it is not a fractal. Gandalf61 (talk) 08:16, 15 May 2009 (UTC)

Aros
I would like to add this to External links. It is freeware. It is an excellent and simple mandelbrot generator/zoomer.

Aros Magic

Any thoughts? --Anna Frodesiak (talk) 09:54, 2 June 2009 (UTC)

Links
And in general, how about Comparison of fractal software — lim 17:28, 9 October 2007 (UTC)
 * XaoS — One of the fastest fractal generators.
 * Wikipedia isn't really a place for software comparisons, unless the comparison is, you know, notable (e.g. the article Windows Vista might include comparisons to its predecessors). I don't think we're in the habit of (basically) reviewing/rating software/games/etc. --Cheeser1 19:22, 9 October 2007 (UTC)
 * I'm talking about something like this: Comparison of container formats, Comparison of audio codecs, Comparison of SSH clients — lim 10:23, 10 October 2007 (UTC)
 * What is about making a list of fractal softwares and small descriptions with how to download, and use them.

Vishvax 06:56, 27 October 2007 (UTC)

See this guideline, which explains what I mean when I say that you can't use the existence of one article to justify the existence of another. I see no reason to make such an article, especially since "fractal software" and the features that define such software are poorly-defined. It's not like an audio codec, which can be compared in some more objective/encyclopedic ways. --Cheeser1 21:22, 28 October 2007 (UTC)
 * Fractal Geometry at Yale - This is Mandelbrot's definitive online resource on the subject. It is probably reasonable to treat it as being in a category of own with regard to relevant links. --Hjb (talk) 16:18, 23 February 2009 (UTC)hjb


 * I know that placing a long list links to fractal generators is discouraged here but I think that the community would benefit from a few good viewers/generators as long as they are vetted by the coordinators. I would like to propose one: http://ladimolnar.com/Fractalia/ BTW, I agree with the fact that a few links like http://classes.yale.edu/fractals/ should also appear here. —Preceding unsigned comment added by 67.160.117.167 (talk) 22:55, 29 August 2009 (UTC)

Self similar?
Not all fractal sets are self similar, some are only self affine. (see for example Faloner's "Fractal Geometry: Mathematical Foundations and Applications".Wilmot1 (talk) 19:44, 1 November 2009 (UTC)


 * The concept of self-similarity includes self affine transformations. The "similariies" in self-similarity can be more general than geometric similarities. Gandalf61 (talk) 09:41, 2 November 2009 (UTC)

Links to other topics
I've linked the text concerning Fractal Art and Fractal Music to their associated articles in the "Applications" section. However this means that Fractal Art is now linked to in three places: in it's own "In Creative Works" section, in the "Applications" section and in the "See Also" section. Surely only one location is necessary? I propose to remove the link from the latter two sections. FatPope (talk) 00:23, 6 March 2010 (UTC)


 * No immediate objections so I've gone ahead with this change FatPope (talk) 09:48, 8 March 2010 (UTC)

Criticalmess (talk) 21:10, 24 March 2010 (UTC) History of Mandelbrot fractal Think this might be useful for putting fractals and their history in perspective - http://classes.yale.edu/Fractals/MandelSet/MandelMonk/MandelMonk.html Apparently, the first occurence of a calculated fractal is earlier than thought. Should this be incorporated somehow? Before editing the article, thought I'd talk this through with the editors here?
 * Seriously? That was an April Fool's joke.  Look it up.TheRingess (talk) 21:22, 24 March 2010 (UTC)

Criticalmess (talk) 19:39, 4 April 2010 (UTC) Ah indeed - my bad -- saw it referenced elsewhere, and no April 1st date was listed. My apologies - nice catch! Nice example of iterated function system animation, trough not for an article: IFS fractal 1 Edo 555 (talk) 13:45, 11 September 2010 (UTC)

Naturally occurring
The romanesco broccoli is not really "naturally occurring" because this vegetable is a product of artificial selection (it is a cultivar of some Brassica species). I'd change it to just natural fractal but that sounds like some mathematical term. And it's such a minutial thing anyway. --137.146.170.254 (talk) 22:24, 16 October 2010 (UTC) Fractal mountains rarely look like real mountains to an experienced geologist or alpinist. Algorithms based on accretion create objects that look like sand piles colored to resemble mountains superficially. I dare say it could be done better if the algorithms paid attention to layers of different hardness and the relative contribution of erosion and different fracture processes. 86.150.232.81 (talk) 21:40, 20 October 2010 (UTC)
 * Well you're right, of course, but I think that's being just a little bit pedantic. Sometimes we can get so wrapped up in being technically correct that we forget we're supposed to be presenting a point to a reader. I'd leave it the way it is. Reyk  YO!  01:33, 17 October 2010 (UTC)

biology
This looks interesting:


 * http://precedings.nature.com/documents/5037/version/1/files/npre20105037-1.pdf
 * (arxiv: http://arxiv.org/abs/1010.4328 )

69.111.192.233 (talk) 14:19, 19 November 2010 (UTC)

Bringing the article up to "code"
This article has fallen off a long way grade-wise over the years, due to increasing standards within wikipedia. Galleries have been verboten on wikipedia for years now, because they are an example of what wikipedia is not. I assigned as many images as I could on the right, per wikipedia standards. This article now has a lead, and I added a few more references. The unreferenced sections are tagged as such. I think I found many of the instances of multiple same wikilinks. Keep in mind that if this article is ever going to pass GAN, let alone FAC, that it needs to be very well referenced, with wikilinks only at the first occurrence of terms within the article. It also needs to be comprehensible to the lay reader, which this article is not at the present time. I'm a scientist and I'm having problems understanding the text. This is the oldest article I've encountered yet on wikipedia, and the most edited, so it would be worthwhile to at least get this article up to Good Article status. Thegreatdr (talk) 14:11, 5 February 2011 (UTC)

Featured article again?
Now that the article has references and has been improved in various ways, perhaps we should reclassify it as a "featured article" again? Soler97 (talk) 09:00, 1 November 2009 (UTC)


 * The article would have to go through the featured article review process first - see WP:FAC for details. Getting an article successfully passed to FA status often involves a lot of work - take a look at Featured article candidates/Euclidean algorithm/archive1 for a fairly recent example. Gandalf61 (talk) 16:12, 1 November 2009 (UTC)
 * This article would have problems with GAN, since there are entire unreferenced sections. They have all been tagged, so you can easily identify them.  Thegreatdr (talk) 14:12, 5 February 2011 (UTC)

How about zooming out?
Fractals always appear to involve zooming in. But how about zooming out? Are there examples of mathematical fractals that can be zoomed in and out infinitely whilst remaining self-similar? Are these still called fractals, or do they have a special name? Greetings Heinzzweinsteiger (talk) 08:53, 27 June 2011 (UTC)

Fractal architecture
A very nice and early (1928-32) example of "fractal architecture" is the Thiepval Memorial in France by Arch. Sir Edwin Lutyens(see http://en.wikipedia.org/wiki/File:Thiepval.jpg). I think this should be added to the "arts section". My english is not good enough to add it myself. 94.110.95.43 (talk) 00:27, 30 June 2011 (UTC)


 * Hmmm. Is there a reliable source that actually refers to this as "fractal architecture" ? Is there any evidence that Lutyens was influenced or inspired by fractals when he designed this memorial ? Gandalf61 (talk) 10:26, 30 June 2011 (UTC)

What is the point of this discussion page?
I'm fed up with people robotically removing things they don't like or that they feel doesn't meet guidelines, whether right or not, with an infinitesimal tag of explanation. This isn't my job. I don't get paid to do this. Entirely undoing something here without a word is like running up to someone writing a nonprofit letter to charity they spent five hours on because the paragraphs were formatted poorly, then walking away saying nothing but "F--- you". That's literally exactly how I feel. How much further do I have to go to investigate ahead of time than taking a chunk of my personal time to thoroughly and exhaustively write out my concerns on a "discussion" page, then waiting over a half-month for no one to answer. Obviously I'm going to try to alter the page to the best of my ability and knowledge if that's all I have. What--is--the--point of a "discussion" page if I can't post my concerns and get a response in half a month... what other processes could I possibly follow before trying to post what I think should be changed.

Experienced people here run to a change like a sudden forest fire, and ignore the kids nagging you on a consistent basis for proper campfire preparation instructions. If you're going to "watch" a page like a vulture, then why not "watch" the discussion page and actually bother helping or discussing something to provide instruction to someone who would like to change something but has questions about it. It's I live in FEAR around here. It's like a horror movie. when someone does this it's like they're literally walking up to my face and spitting in it.

I wrote the section "Ghastly definition" above. I thoughtfully, thoroughly expressed a concern about the page in a proper place. No one responded for 16 days so I just did the best I could to make the change. It gets ripped down in practically nanoseconds for being a "POV essay", when I thought I'd written the most NPOV and Wikipedia-friendly thing I've ever written. Assuming I'm wrong, why not suggest improvements, or god forbid click the secret button to discover where I actually asked the question 16 days ago, and respond to that with suggestions. This just shouldn't even be ALLOWED. If you're going to "vulture" an article, you should be required to "vulture" the discussion page.

I'm sorry for expressing my anger around anyone to whom it doesn't apply, but you can't imagine what it's like to have a level of ADHD where it can take you half a DAY to write one long email or a couple paragraphs of an article or essay. If I take my time to viciously adhere to the rules and guidelines as well as I'm able and as far as I know them, I'm sure as hell gonna be infuriated if that nonprofit time I'm taking of my own life is spat on in ways I'm too tired to even describe anymore.

Would someone please tell me how you're supposed to reach a consensus around here? If my alteration just now was a "POV essay", then how does anybody go about editing anything unless there's a reference number after every..single...word in every article. What I wrote was very general. How do you provide references for statements like "widely used across science and art"? In retrospect, after realizing afterwards that there was a page specifically for fractal art, distinguishing "fractal" from "fractal art", my concern and edit would have been different, but that's still something that someone could have explained weeks ago when I went through my concerns thoroughly.

Minus my previous concerns that there weren't any explanation of fractal art around here, my criticism of the main definition still stands. My paragraph still stands as a suggestion preface, though I don't know how it could possibly be edited or modified or improved, without someone editing or modifying or improving it... Squish7 (talk) 10:58, 3 October 2011 (UTC)


 * For reference, this was the paragraph I added at the start of the article that got removed: "The term fractal is of strictly mathematical origin, yet as computer processing speeds increase and allow greater resolutions and exploration, the term continues to expand and shift in meaning, branching into, or even creating, entire applications not originally foreseen. The term has been used in regards to many various corners of science and art (e.g. geometry, probability, meteorology, visual art) and across many different mediums (imagery, video, audio, film, the web, stock art, etc.). As the term evolves into versatility, so does the fuzziness grow of what precisely the term means, or should mean. The easiest way to unravel the concept, then, is to begin with its strict mathematical roots."
 * Instead of changing the definition, I tried the lightest thing I could think of to address my issue: prefacing the article with a note that the main definition given is strictly a mathematical one. I still think it's a decently ghastly one (somewhat less so now that I see that Wikipedia has segregated "fractal" and "fractal art"), but I didn't change it.  I thought arranging viciously obvious, widely known, and verifiable facts to flag the narrow/poor definition was behaving properly.  If you disagree it's a poor definition, then SPEAK. UP. Squish7 (talk) 11:13, 3 October 2011 (UTC)


 * I removed your new paragraph from the article's lead for two main reasons. Firstly, the paragraph was written from your personal point of view, without any supporting references or sources. Secondly, the lead section in a Wikipedia article is supposed to summarise the contents of the article, whereas your paragraph took the lead off in an entirely new direction that is not supported by the rest of the article. If you have sources that say that the definition of the term "fractal" is changing or shifting as you claim, then I suggest you cite these in a section in the body of the article, rather than modifying the lead section. Gandalf61 (talk) 11:47, 3 October 2011 (UTC)


 * It was not written from a personal point of view; it's just about exactly what would be there IF the article on "fractal art" did not exist (information that would otherwise be here), which I didn't know about. If there was not that page, my paragraph would have been a very appropriate addition/correction to what was here, as this article makes extremely little mention of fractal art.  I couldn't have known that W put all the artistic elements in another article.  That's why I posted a discussion item and waited over half a month, so someone could correct me if there was something I was missing.  If I post something akin to "this article is completely wrong", and get no response, why wouldn't I assume that the article is very out of date, and go correct it based on what I know, modifying with the unarguable, verifiable basics.


 * My issue isn't that you removed the paragraph, my issue is that you wasted my time by not monitoring the page and offering an answer when I all but directly STATED that I was going to do that. I even brought UP your concern specifically! (I.e. that my feelings on the matter may be prone to issues of original research, hence the ASKING of the question before I bothered doing anything.)  That is, there's a decent chance if I think there's an error or strong omission on the page, that I'm working on a way to fix it, or a special article to add.  At the absolute least you could have bothered to look at the discussion page and responded to my question WHEN you removed my paragraph, since obviously I'm going to want to know/ask why it was removed, even if it would have been too late to save me from wasting my efforts.


 * My anger is from that I used my cumulative knowledge to behave in a manner that was very appropriate, and got treated as if I logged on for my first time and just started throwing my opinion up without any regard for the policies or method of behavior. Anybody in general who is not specifically trying to behave improperly is going to feel stepped when an edit is removed without a full sentence of explanation.  In this case, I took extra efforts to inquire beforehand, and waiting a good time for a response.  Do I really have to put "This is my precise revision that I will paste in two weeks if I do not get a response to this post"?  Of course my paragraph changed the nature of the article; if a discussion page is not monitored, why would I assume that the article is current and accurate? Squish7 (talk) 21:33, 7 October 2011 (UTC)


 * FYI, I think the definition--even outside the artistic issues now covered--is still very poor. It's completely disprovable, as I've done, above.  Just the fact that it references someone else's precise quote is bad news; there's an element of plagiarism, there, in fact, just as there would be if you just pasted the article where that quote came from, saying "this is what a fractal is about, because it was written by an expert". Ughgh. Squish7 (talk) 21:44, 7 October 2011 (UTC)


 * Okay. I hope you are feeling less angry now. Gandalf61 (talk) 10:24, 9 October 2011 (UTC)

Error in fractal animation
The frames of this animation flip horizontally after the second frame and it makes things confusing (and inaccurate). It's a great animation otherwise, but I don't know how to fix it. I left a note on the creator's page, but he has not been active recently. I hope someone can make it right. AmateurEditor (talk) 04:02, 20 November 2011 (UTC)


 * The animation has been fixed/reverted. Big thanks to Tó campos. AmateurEditor (talk) 01:24, 30 December 2011 (UTC)

Inventor of fractals at Fractal art
Could watchers of this article please contribute to the discussion at Talk:Fractal art. Apparently if you don't like the addition of improperly sourced claims that Kalantari invented fractals in 2002 you have an agenda. - Shiftchange (talk) 03:03, 11 June 2012 (UTC)

Generation / Beginner's Help
I want to add, for anyone who does not understand this topic, that a FRACTAL IS:

The 'SET' (look up Mathematical sets) of all the Points, on the COMPLEX PLANE (look up complex numbers) which when iterated through a fractal equation, does NOT escape to infinity. (For example, the Mandelbrot is, z(n+1) = z(n)*z(n) + c Where 'c' is the point you are testing......).

If this is any help, please say so!

74.199.8.90 (talk) 22:45, 8 August 2009 (UTC)
 * No, I'm afraid that is not helpful. The sets you describe are only one example of fractals. The definition in the article is good.--seberle (talk) 17:59, 12 August 2009 (UTC)


 * I'm thinking I meant more of a specific ALGORITHM explanation, how to actually visually generate a fractal. Mostly, because this does not exist in articles about specific fractals.  —Preceding unsigned comment added by 74.199.8.90 (talk) 21:33, 27 October 2009 (UTC)


 * Fractals are not necessarily in the complex plane, or complex hyperspace for that matter. The escape to infinity test applies to Julia and Mandelbrot sets, but not much else.  By that definition, clouds, mountains, and coastlines aren't fractal; there are many other real-plane and real-space examples.  The formal definition is usually given as "a set of points whose Hausdorff-Besicovich dimension strictly exceeds the topological dimension," but that excludes the Hilbert curve so it's all kind of up in the air.


 * There are many algorithms that generate fractals and new ones are invented all the time. It's easy to cite any particular one, but the algorithm to use depends on what you want to see.  Most are amazingly simple -- the core of a Mandelbrot set generator is perhaps three lines of C++ -- and that's part of the beauty.  The wikipedia is full of generation algorithms, simply search for what you want. Diffusion limited aggregation and the Sierpinski triangle are good places to start.  — Preceding unsigned comment added by 66.219.227.172 (talk) 04:52, 19 September 2012 (UTC)

Differentiability
Hello everybody Seeing as fractals are - at least - a bit difficult to understand for many non-specialists, I am very much a proponent of making this article seem less cryptic. Seeing as there has been a lot of work done this far, might I continue the process by making a suggestion? :-) Concerning differentiation, some readers (stumbling on this article) may be familiar with the idea of differentiable functions, and they may also be familiar with the idea that a given function can be non-differentiable at either one point or many points. Remember, differentiability is - so to say - the effect of both continuity and congruency; for a function to be differentiable in a certain point (or over a certain interval), it must be both continuous and congruent in that point (or over that interval)... What could possibly be very or slightly confusing to the casual reader, might be the description of fractals as being "continuous but not differentiable". May I suggest that the description be expanded to "always continuous but never congruent, and therefore not differentiable"? On the one hand, I imagine that a reader, who already knows something about differentiable functions, may be surprised, this situation ought to be addressed... On the other hand, a reader who knows nothing about differentiation at all, may be confused... So, is it okay, to make the proposed expansion? What is your opinion?Jaloqin (talk) 22:20, 21 May 2013 (UTC)


 * IMO "continuous but not differentiable" seems clear enough, but feel free to try and include congruence and see how it works out, seems like a more familiar term (assuming you refer to the geometric meaning). Nice suggestion! M&and;Ŝc2ħεИτlk 23:04, 21 May 2013 (UTC)
 * I don't see what "congruence" means in this context. — Arthur Rubin  (talk) 00:52, 22 May 2013 (UTC)
 * I've looked around other articles on Wikipedia; and maybe the notion of congruence would confuse the matter slightly more. In this context "congruence" would have the connotation of "smooth" as opposed to "fractured", whence comes "fractal". So now I'm not sure... Let me think about this. — Preceding unsigned comment added by Jaloqin (talk • contribs) 05:42, 22 May 2013 (UTC)

Symmetry
Not once symmetry is mentioned in the whole article. Isn't Fractals a symmetric type of pattern? "Patterned Self-Similarity" I only added "See Also" for symmetry. Like the above comment of Differentiability suggesting to make the article less cryptic, how about talking symmetry of self pattered similarity? Symmetry is a word most readers can relate and understand. Neoking (talk) 00:28, 1 June 2013 (UTC)

Spoken Article
Why is there a spoken version of this article from 2005? Is there any real use of that, considering how outdated and incomplete the information must be? Exercisephys (talk) 16:16, 16 June 2013 (UTC)

Merger proposal
I propose that Non-Euclidian_geometry be merged into Fractal. I think that the content in the Non-Euclidian_geometry article can easily be explained in the context of Fractal, and the Fractal article is of a reasonable size that the merging of Non-Euclidian_geometry will not cause any problems as far as article size or undue weight is concerned. Or vice-versa. 187.12.26.206 (talk) 17:56, 25 June 2013 (UTC)


 * No chance. Non-Euclidean geometry is not necessarily fractal and the two topics are separately different things, worthy and notable for separate articles, and the articles are already large. I don't find your reasons at all persuasive. M&and;Ŝc2ħεИτlk 18:01, 25 June 2013 (UTC)
 * Oppose, and I don't really see how one who understands both concepts would think that. — Arthur Rubin  (talk) 18:57, 25 June 2013 (UTC)
 * Oppose -sorry, but they really are are completely separate topics. Gandalf61 (talk) 19:57, 25 June 2013 (UTC)
 * Oppose And those two concepts have very little in common.Prokofiev2 (talk) 12:32, 26 June 2013 (UTC)


 * [[Image:Symbol confirmed.svg|20px|link=|alt=]] . Have added some strikes to the first request. 177.135.4.98 (talk) 22:26, 5 July 2013 (UTC)

Self-Affinity
Can anyone provide some pictures to the self-affinity article?Lbertolotti (talk) 00:06, 31 July 2013 (UTC)

Fractal Geometry
Fractal geometry redirects to Fractal. The page Fractal should therefore constitute an introduction to fractal geometry. Does anyone mind if I begin to re-edit this page to repackage the existing text, info and concepts in a more orderly and readable style? JackWoehr (talk) 17:10, 4 January 2014 (UTC)
 * If you look at the Talk page, you will see that people have been asking for a more orderly and readable style for quite some time. If you have good ideas for editing this page, especially the opening, go for it. --seberle (talk) 20:15, 8 January 2014 (UTC)
 * And should the redirection be reversed? JackWoehr (talk) 17:15, 4 January 2014 (UTC)
 * I'll give it a shot, thanks. JackWoehr (talk) 14:20, 9 January 2014 (UTC)
 * This is requiring a lot of remedial study! You can watch FractalNet HD - Slow deep Mandelbrot zoom while you're waiting for me to start editing :) JackWoehr (talk) 05:26, 11 January 2014 (UTC)

Introduction: Where is Figure 2
Figure 2, presumably being a curved line divided into 4 pieces, is referenced in the article, but doesn't actually appear in the article. 135.23.85.212 (talk) 15:25, 3 January 2015 (UTC)
 * Looking at the article history, Figure 2 was renamed to figure 2a at some point. It might also be helpful to define the use of curve in the introduction, since one might naturally assume the article means a smooth curve. 135.23.85.212 (talk) 15:35, 3 January 2015 (UTC)

verry little math
Altough most Wiki pages about math subjects are unreadable because of the nathlanguage used. This article contains no formula at al. Well that's great its a good read. But perhaps a little bit about the mathematics behind fractals just at a level one can create his own basic version fractal would be nice. —Preceding unsigned comment added by 82.217.143.153 (talk) 16:48, 23 February 2008 (UTC)

I have included a small article about how I recently discovered how to hand draw fractals of anything. I left links showing where to go to see a demonstration. I would like to download a step by step process (in diagrams and an explanation) in which I can show any one (with out using complicated equations), how to hand draw a fractal of any shape. Any help that you could give on how I can download images would be great.--JASONQUANTUM1 (talk) 17:41, 7 July 2008 (UTC)

Yeah I agree about putting more math in this article, I am a student of mathematics and find fractals very interesting. I would really like to know about the math behind a fractal.

Missing from the article seems to be the information that the Koch snowflake is actually Three Koch curves fitted together. Also the Fractal Dimension is on a separate page from the main article. I would say that the importance of fractals comes into play when we try to fit Euclidian geometry onto shapes that are far from Euclidian in nature. Because these shapes are so complex we need to study them in different ways from the Euclidian shapes. Just as there are Euclidian like shapes in nature also we find Fractal like shapes and Euclidian geometry will not help us to understand them. Richardjames13 (talk) 11:01, 11 January 2009 (UTC)


 * I agree. I think that there definitely could be some more maths in this article.  For example, are there equations to represent fractals?  Or to find their area?My 2 Cents&#39; Worth (talk) 17:20, 22 May 2010 (UTC)

If I may suggest just borrow stuff from BM "The Fractal Geometry of Nature", there's plenty of math there to drawn from. --Lbertolotti (talk) 19:54, 19 February 2013 (UTC)

There is an equation for self similar fractals in being able to calculate the dimension of the set. This is done by considering what happens at each step, that say for example you have a square and on the next step you draw a line through and divide into 4, making 4 copies each 2 times smaller than the original square then the dimension of this set can be calculated from d = ln(n)/ln(r) where n is the number of copies and r is how much smaller they are. So in the case of this square, we have d = ln(4)/ln(2) = 2, which should be expected as a square (or grid) is 2D; in fractal cases d is a fraction. — Preceding unsigned comment added by 130.88.174.207 (talk) 20:21, 14 January 2015 (UTC)

iterations on complex functions till break
the unity circle lenght 1 diverges somehow on the counter variable

the forula onc fashion gets lost like the others

once children used it to test the preformance of the arithmetical logical unit of a processor now they seem outdate - stange the is no info out there79.234.234.220 (talk) 11:24, 19 January 2015 (UTC)

Point for immediate action
The numbered Figure approach (Fig. 1, 2, 3) in this article needs to go, in favor of an open structure. This is clear from wikipedia guidelines and common sense, and imposing it only has the impact of making article editing—and so long term evolution toward the highest quality article—more difficult, e.g., saddling an incoming editor that wants to add a Figure with the tedious and unnecessary task of renumbering downstream of their edit. (And the next addition the same, and the next.) Given that in areas of modern research, where knowledge grows, there is reason to expect accrual (addition) of information, this structure is just silly to hang on to. All the more since we use open source images, where the likelihood that a better or additional image might become available is high.

I have no desire to make this editorial decision at this article, where I come as a non-mathematician (though applied scientist using maths). But it needs to be done, and soon. But regular contributors should. Its remaining cannot be that you wish to control the article, and stop it from changing, I know (for this is contrary to the spirit of WP). No reason, then, to give that appearance. Le Prof. 71.239.87.100 (talk) 19:10, 9 March 2015 (UTC)

Second point for immediate action
The lede of this article is ill-served by opening with the Menger Sponge example, first because of its complexity, second because its picture does not appear. This design of presentation is "backward," pedagogically. Develop the article lede, carefully, from less to more complex wording, concepts, and images. In this vein, consider using a simple image with iterations, of the sort that is done with to a Koch curve, to open the article. (The Sponge mention needs to come later, not in the opening of the lede, and wherever it is put, it should not be without an image.)

Bottom line, it's nonsensical that one comes to an article, and then in the first three sentences is asked to leave the article to acquire basic understanding of the article subject (in this case, by following the Sponge wikilink); this is bad teaching/encyclopedic writing practice. (Jumping to the Menger Sponge article makes this all the more clear, because that article opens in defining it as a 3D generalization of the Sierpinski carpet, which is in turn a 2D evolution of the Cantor set; bottom line, you cannot begin to understand the Fractal article opening sentences without going several wikilinks deep, 3-4 steps away from this article to begin to understand it!)

Shown at left and right, here are examples of things much more simple, the likes of which (I repeat, the likes of which) should appear instead of the complex, 3D Sponge example. One option to open the article that has significant pedagogic value is the the animation for the Koch curve. A second is the Sierpinski carpet animation, though somewhat obnoxiously coloured (Go U Northwestern, ).

Please, make a change to this opening example given, and show an image to go with the simple example—so the article is a useful teaching venue for students, and not something where mathematicians just talk to themselves. (We already have Wolfram for that.) Le Prof.  71.239.87.100 (talk) 19:10, 9 March 2015 (UTC)

Fleas
Would there be a place in the article, perhaps History, for this ancient verbal fractal? (based on Swift, 1733):
 * Big fleas have little fleas,
 * Upon their backs to bite 'em,
 * And little fleas have lesser fleas,
 * and so, ad infinitum.

-- Sarlo (talk) 16:17, 14 March 2015 (UTC)


 * I would be in favor, backed by suitable sources. It is maybe a pre-fractal concept. Jonpatterns (talk) 11:03, 15 March 2015 (UTC)

Anon random edits
There are several errors and (probably intentional) placement of unimportant material before important material made by an IP6 anon. Could he explain why he thinks that Mandlebrot is more important than the definition? — Arthur Rubin (talk) 00:47, 21 September 2015 (UTC)
 * Specifics:
 * Mandlebrot's "definition" of the concept should not be in the first paragraph of the lead. I'm not sure it should be as far down as it is in the stable section, but it should not be first.
 * The "Creation" section is the IP's opinion, unsupported by facts. It purports to summarize the following specific construction techniques, but fails to do so.
 * — Arthur Rubin (talk) 01:24, 21 September 2015 (UTC)

Hilbert Fractals and the Wilczekian Grid (Hilbert curve and more)
Frank Wilczek wants to evolve Hilbert space Fractals in order to update SUSY. The main article doesn't mention engough what Hilbert Fractals are!

also Hilbert curves are not mentioned extently! Why? Read that wikipedia article, they are Hilbert fractals, but they can evolve more in the future as a field of study.

( enrica (talk) 12:04, 20 October 2015 (UTC) )

Additional animated example
I made calculations for one of the most beautiful fractal I have seen:



I think this example is attractive and it's a good way to show repetitive structure in animation. Though the main article Ana's Fractal is not sourced, it can not possibly be sourced, as there is no source. I found and named it: this can not be checked directly, but anyone can search for other sources of this fractal, and if there is some other older source with a different name, it will be changed. — Preceding unsigned comment added by Guggger (talk • contribs) 04:28, 27 August 2016 (UTC)

Controversy with Fractal Authentication of Jackson Pollock's Art
The "In Creative Works" section of this article excludes the controversy surrounding the use of fractal authentication for Jackson Pollock's unsigned paintings which has been clearly documented in several articles here and here. The Jackson Pollock page at least mentions this controversy here and has updated citations expanding on multiple feature analysis to support authentication. I would suggest to contributors on both articles that a more complete and consistent story of on-going controversy be documented in both articles. If neutral editors are curious why there is even a controversy to begin with, the stakes of being able to authenticate unsigned Jackson Pollock paintings is described here. Pdruley (talk) 15:47, 28 September 2016 (UTC)
 * If something is missing please be bold and add.Jonpatterns (talk) 11:14, 29 September 2016 (UTC)

Should history section mention Benoît Mandelbrot
Should history section mention Benoît Mandelbrot? Jonpatterns (talk) 11:14, 29 September 2016 (UTC)<


 * The third paragraph is about Mandelbrot. Feel free to add any missing important details. --seberle (talk) 15:39, 2 October 2016 (UTC)

Opening sentence
"A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension." This is entirely inadequate as a mathematical definition. "Usually?"--seberle (talk) 05:24, 31 August 2013 (UTC)


 * The opening sentence is not meant to be a mathematical definition. That comes further on in the article, in the Characteristics section - in fact, there are several definitions, which are not entirely equivalent. The weasel word "usually" is probably a nod to space filling curves such as the Hilbert curve, which are often classified as fractals even though their fractal dimension equals their topological dimension. Gandalf61 (talk) 07:38, 31 August 2013 (UTC)


 * Even if it is meant to be an ordinary definition (rather than a mathematical one), it is very odd and rather confusing. Whatever the opening sentence is meant to be, it reads as an introductory definition and fails to be meaningful because of the weasel words "usually" and "may". Some sort of definition is needed at the beginning before going into what fractals "usually" are or what they "may" be. Also, the reference at the end of the sentence, which anyone seeking clarification would click, contradicts the word "usually" with this citation "A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension". An opening sentence does not have to be a mathematical definition, but it should be a clear introduction all the same. My recommendation would be to start with a proper definition based on Mandelbrot's citation and then follow it up in the succeeding sentences with caveats about how this basic definition has been extended to include other mathematical objects. seberle (talk) 15:18, 2 September 2013 (UTC)

I can say this much, the opening sentence, "A fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension[1] and may fall between the integers." is nigh incomprehensible to a non-mathematician like myself. Isn't it a textbook bad definition of something to include that word in its own definition (i.e., "fractal dimension" in the lead for an article entitled "fractal")? RobertM525 (talk) 00:28, 10 December 2013 (UTC)
 * I think it is a very poor introductory sentence for a Wikipedia article, whether mathematician or not. seberle (talk) 19:13, 10 December 2013 (UTC)


 * One noticeable feature of fractals, to everyone, is their self-similarity. Since this thread is about the first sentence, maybe a slight permutation of sentence would fix things? I'll try it. M&and;Ŝc2ħεИτlk 09:47, 11 December 2013 (UTC)


 * Some fractals are self-similar. AFAICT, all artificially created fractals are, while no naturally occurring objects appear self-similar across more than a narrow band of scaling, yet all all natural objects appear to be non-differentiable, lacking any Euclidean "point" per se. Look closely enough, and everything gets rough and chaotic.BurntSynapse (talk) 18:46, 6 May 2017 (UTC)

The first sentence currently reads, "A fractal is a mathematical set described by fractal Geometry, the study of figures exhibiting fractal dimension." I understand that this is a difficult topic to define, but this sentence conveys no useful information to anyone who needs a definition of fractal! Rightoncommander (talk) 18:51, 2 February 2014 (UTC) 18:50, 2 February 2014 (UTC)
 * I did the last major edit on that ... The problem is, why is there a Fractal page and no independent Fractal_geometry page? Then it wouldn't be circular. As things are currently, we have to commence with a quick reference to fractal geometry to be able to talk about fractals. I am not sufficiently expert a Wikipedia editor to create the independent Fractal Geometry page but if some more expert editor would create it, I would help partition the subject matter between the two pages. I assert that this is the easiest way to address your valid criticism. JackWoehr (talk) 03:55, 3 February 2014 (UTC)
 * I have reverted to a previous version of the opening sentence which does not mention fractal geometry, but instead gives a more informative definition in terms of self-similarity at differenr scales. Gandalf61 (talk) 10:13, 3 February 2014 (UTC)

External links modified
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External links modified (January 2018)
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Comment
Utter gibberish. In no way exemplary of how an encyclopedia should read. — Preceding unsigned comment added by 2601:CC:0:E303:6187:87:EC44:6C51 (talk) 01:40, 12 January 2019 (UTC)

Eh? This makes no sense.
> Fractals are encountered ubiquitously in nature due to their tendency to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot set.

How can the ubiquity of fractals in nature be illustrated by the Mandelbrot set? The Mandelbrot set is not part of nature.

How does the fact that fractals tend to appear nearly the same at different levels account for their ubiquity in nature?

What is a "level", anyway?

Is it even true that fractals are 'ubiquitous' in nature? I mean - a tree trunk looks different to a tree branch, right? It's not fractal at all!

203.13.3.89 (talk) 02:06, 4 March 2019 (UTC)

> In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension.

What about noneucledian spaces? You can totally have fractals in hyperbolic space!

> As mathematical equations, fractals are usually nowhere differentiable.

Can't quite put my finger on it, but this doesn't make sense either. Fractals aren't (necessarily) equations, and it isn't equations that are differentiable: it's the things that equations describe. I mean, the Mandelbrot set isn't an *equation* ('this' equals 'that'). 203.13.3.93 (talk) 02:13, 4 March 2019 (UTC)

> The mathematical roots of fractals have been traced throughout the years as a formal path of published works,

To say that fractals have mathematical roots sort of implies that they used to be mathematical, but no longer are (eg: "Germanic roots of many English words"). Which is not the case.

"have been traced" - by who?

"a formal path of published works" - well, duh, that's how mathematics gets done. Of course somebody bloody published it! If we are talking about a timeline, it should be done as a table or something, not as a story. Oh, and is that supposed to be "formal path of published works" or "path of formally published works"? Because if it's mean to be the former, I don't know that it's true at all. As far as I know, no-one is the formal custodian of mathematical history, whose job it is to curate all the pathways between mathematical works.

A the previous comment on this talk page points out, this article is just awful. 203.13.3.90 (talk) 02:23, 4 March 2019 (UTC)


 * I'm sorry that you feel this way and you are of course correct for a number of these examples. Let me address some of them:

How can the ubiquity of fractals in nature be illustrated by the Mandelbrot set? The Mandelbrot set is not part of nature. How does the fact that fractals tend to appear nearly the same at different levels account for their ubiquity in nature? What is a "level", anyway?
 * Although the phrasing is bad, the part about the mandelbrot set is supposed to provide an example about the increasingly small detail in the second part, not the part about nature. It should be separated into 2 sentences. A "level" is essentially the zoom and detail of the fractal in successive similar-looking steps.

The mathematical roots of fractals have been traced throughout the years as a formal path of published works,
 * You are right; this sentence is just about completely useless and I would support removing it. I would encourage you to make an account and try improving the page; its people like you who could possibly improve this kind of page - Integral Pythonclick here to argue with me 12:41, 4 March 2019 (UTC)

the binary time format is fractal in nature
http://anthony.liekens.net/index.php/Misc/TrueBinaryTime --Backinstadiums (talk) 16:47, 20 July 2019 (UTC)
 * False, even if anyone should care. — Arthur Rubin  (talk) 18:16, 21 July 2019 (UTC)

Is the area of these shapes infinite too?
So apparently I know that these shapes' perimeter are infinite, but is the area infinite too? — Preceding unsigned comment added by 2600:1700:3A20:25B0:806F:4195:A280:573F (talk) 18:26, 21 July 2019 (UTC)
 * $Begin original research$A bounded plane figure which has an area, has a finite area.$End original research$ — Arthur Rubin (talk) 06:35, 22 July 2019 (UTC)

Suggestion
Hopefully this can offer an eventual merge? Sr13 05:48, 20 June 2007 (UTC) Adding a reference for item 34 on the achieved talk page (page2). A reference from the Nexus Journal: Daniele Capo, The Fractal Nature of the Architectural Orders.  Nikhil Varma 14:23, 21 June 2007 (UTC) Adding a link to www.fractalguide.com —The preceding unsigned comment was added by 149.254.192.192 (talk • contribs).

I have recently developed a web based fractal viewer. Does anyone think a link to the viewer on the fractal pagewoulds be useful. is where it can be found. —Preceding unsigned comment added by 199.44.137.1 (talk) 16:41, 9 October 2007 (UTC)


 * See WP:EL. A new link is not necessary.TheRingess (talk) 16:50, 9 October 2007 (UTC)

Thanks for your opinion —Preceding unsigned comment added by 199.44.137.1 (talk) 19:07, 9 October 2007 (UTC)


 * There is nothing that would rule out that link in WP:EL, in fact items 3 and 4 of what should be included fit the given site well, and it passes all tests of what not to include. EL is also a style guide, not a strict policy. All that said, I don't like the link simply because of the ugly layout and palette used. Fix that up, and the site would make a great addition to this page. Nazlfrag (talk) 06:28, 31 December 2007 (UTC)

This page is too complicated. This is an encyclopedia for lay people to gain knowledge. You need to explain simply what a fractal is and then get more complicated. As you get more complicated, you should try to add words to basically explain what you're saying in non-technical terms. As fractals have implications in the inflation model of cosmology, people want to understand this, but this page is nearly incomprehensible without a good understanding of math. Lose your ego and explain! PeterB1517 (talk) 17:15, 3 July 2019 (UTC)

I've said this on a few pages, science ones especially. We need a layman's section for, well, laymans like me, and young children. Say a young person learning about fractal comes here to learn some more, and come across the first sentence 'In mathematics, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension.' I know there are hyperlinks to the terms used, but if that kid needs just a simplistic explanation of the subject here, hes not gonna get one. The poor kids gonna have to click on every hyperlink for every terminology used like, self-similar subset, Euclidean space, fractal dimension, topological dimension. Etc, before he can understand what hes reading, so this is basically just a request for simplistic/layman sections on all scientific, mathematical etc wikipedia pages. D0S81 (talk) 17:25, 12 April 2020 (UTC)
 * That would agree with the Manual of Style. See MOS:JARGON and WP:ONEDOWN. AmateurEditor (talk) 18:49, 12 April 2020 (UTC)

Lead sentence
This article has wrestled with the lead sentence for quite some time. Fractals are not easy to define.

Currently the lead sentence reads: "In mathematics, a fractal is a self-similar] or not] subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension." It is displayed exactly like that, brackets and all. I am guessing someone tried to make a link for "self-similar" and failed. I am not sure how the "or not" is supposed to display, so I cannot correct the sentence. --[[User:Seberle|seberle (talk) 09:24, 10 February 2021 (UTC)

hi
i like fractals 1fractal4 (talk) 16:57, 16 April 2021 (UTC)

Proposal
I propose expanding List of fractals to include more information so that it looks like this and then moving the page to Table of fractals. Thoughts? Please respond here. &#x27A7; datumizer   &#9742;  19:08, 5 July 2021 (UTC)

Adding the cardinality of aleph-two for the set of deterministic fractals in R^n
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. It shows the set of fractals in R^n has a cardinality of aleph-two(beth-two). 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) 01:47, 9 July 2021 (UTC)

Smaller scales?
What are 'increasingly smaller scales'? (1st para.)--Damorbel (talk) 15:25, 24 November 2021 (UTC)