Talk:Fractal/Archive 1

Untitled
I removed the following. Who exactly believes this? -- Miguel

Fractals are believed to hold the key to the design of life itself. Fractals find their way into all living structures on earth, and thus are an abridged creation of organic life itself.


 * Poorly phrased, but with a grain of truth. Fractals are all over the natural world because self-similarity is a good way to build large structures from a simple set of (genetic) instructions. The same is true, to a lesser extent, of some non-biological structures --basically anywhere there are non-linearities.


 * Urhixidur 22:12, 2004 Aug 20 (UTC)

-

The text below is useful for dates and names, but does not provide a good (or even accurate) description of what a fractal is, so I hope the mathematicians here can do better. I wrote an article on the subject once for a graphics magazine here, but it's a layman's introduction, not very encyclopedic (but feel free to steal from it). --Lee Daniel Crocker

Fractals A visual graph of an equation involving the repetition, or reiteration, of a geometric figure. The general form is: f(z) = z^2 + c     This can graph the Mandelbrot set, named after Benoit B. Mandelbrot, (born 1924), who first used a computer to graph these number sets. Also of note is Gaston Julia, (born 1893), who first studied iterative transformations, of the form: x^2 = x^2 + a and y^2 = 2xy + b, the basis for Julia sets. Their work has become the basis of fractal art. Colored number sets, displayed on computer screens, or printed on paper, reppresent some of the most beautiful, compelling images you will ever see. Many Fractal Art Galleries can be found on the internet, and are well worth a visit. gg

---

I personally think you should Wiki your article. Encyclopedias need the general and the specific. And if the Fractint documentation is Public Domain (my bad. It isn't. But we could ask...) then that can be the start for individual entries on each fractal type. Just my £0.02 worth... Dave McKee

Mandelbrot and Richardson?
I'm confused by the content of this article. Who exactly wrote "How Long is the Coast of Britain?" From the last para in "History", it seems like it might be Mandelbrot, but the next section only mentions Richardson. However, almost all of the sources I find on google when I search for "how long is the coast of britain" report Mandelbrot as the one who asked the question. Can this be clarified somewhat in the article? - DropDeadGorgias (talk) 13:41, Jul 7, 2004 (UTC)


 * Richardson did the research and collected the statistics but the paper How Long is the Coast of Britain? was written by Mandelbrot. I have fixed this in the article. Gandalf61 16:03, Jul 7, 2004 (UTC)

(Contested -- Jul 2) Fractal
I noticed we don't have many featured articles on mathematics. This is one of the few math topics with widespread appeal and lots of pretty pictures. Although it looks a bit short at first, if you follow the links to the specific types of fractals, there's a lot of material. I've verified the image copyrights and they all appear to be GFDL or public domain. --Shibboleth 23:56, 2 Jul 2004 (UTC)
 * Support. Having spent hours upon hour exploring microscopic regions of the Mandelbrot set, I can only say, "It's about time..." Denni ☯ 05:49, 2004 Jul 3 (UTC)
 * (Not a vote). It looks like all of the images are in the public domain (created by Wikipedians), but some fail to note so explicitly. Jeronimo 11:31, 3 Jul 2004 (UTC)
 * Now the only picture that does not explicitly specify public domain or GFDL is the Koch snowflake picture. Unfortunately the user who added it is not around anymore, so we cannot ask him to add the label.  But I am inclined to think an elementary mathematical illustration like this cannot be copyrighted anyway.  If it really bothers someone, it should be easy to re-create. --Shibboleth 21:38, 4 Jul 2004 (UTC)


 * Object. 1) The lead section needs to be longer, including a mention that "fractal"s is derived have "fractional dimension", amongst other things. 2) We should discuss Fractal art in this article to some extent, given that most laymen encounter fractals in that form. 3) Similarly there needs to be a much more in-depth discussion of applications: Fractal landscape, Fractal compression...; we currently have "Fractal techniques have also been employed in fractal image compression, as well as a variety of scientific disciplines." 4) The article emphasises self-similarity; an image sequence would be one obvious way to illustrate this concept, e.g. a sequence of zooms on a fractal, showing self-similarity on different scales. 5) You can't read much about fractals without bumping into a discussion of chaos theory; what's the link? The article doesn't discuss this. &mdash; Matt 16:39, 3 Jul 2004 (UTC)
 * [Etymology of fractal: although some web sources say fractal is derived from "fractional dimension", Mandelbrot himself says he coined it from the Latin fractus - see, for example, Mandelbrot's essay "Fractals - a geometry of nature" in "The New Scientist Guide to Chaos". I have added this to the article.Gandalf61 11:41, Jul 4, 2004 (UTC)]
 * Ah, great. Maybe it's worth mentioning the mis-etymology of "fractal" as well, since it's quite common? &mdash; Matt 15:36, 4 Jul 2004 (UTC)
 * I've added a sequence of images demonstrating self-similarity. [[User:Sverdrup|❝Sverdrup❞ ]] 13:54, 4 Jul 2004 (UTC)
 * Thanks, spot on. &mdash; Matt 15:36, 4 Jul 2004 (UTC)
 * This is good but I don't find the sequence of images quite clear enough as it is. The problem is that since they all look so similar, it's hard to understand at first sight that the fourth image is a 100x (or whatever) magnification of the first: it looked to me at first like just a big rotated chunk of the first, until I understood that your red squares meant "zoom".  I would like a label added to the top of each image saying "Magnification: 1x", "Magnification: 5x", "Magnification: 25x" (or whatever the exact numbers are).  Or even better, nice arrows linking the red square of one image with the next image.  Though come to think of it I don't really know which would work better. --Shibboleth 21:59, 4 Jul 2004 (UTC)


 * Support Avala 19:34, 4 Jul 2004 (UTC)
 * Neutral - the article looks fantastic and reads fine but it does seem to be a bit short. I think some basic expansion is needed, but since I don't know much about this topic, I don't know what to suggest (and thus my non-vote). :( --mav 10:11, 10 Jul 2004 (UTC)

Introduction
The use of self-similar in the introduction as a definition of fractal, which has now been edited out, was a compromise intended to reconcile what mathematicians know the true definition to be, and what in practice the usage is amongst physicists and others. This distinction has now been lost, really.

Charles Matthews 06:25, 13 Jul 2004 (UTC)


 * The problem I had with the way it was originally phrased is that there is no clear idea of what "self-similar" means by itself, even for physicists or other people. Many people (scientists included) may think that "self-similar" means what is really called "exactly self-similar", and may not realise that this word is being used to stand in for a variety of types of self-similarity. Just putting "self-similar" without a caveat or description about different forms of self-similarity seems misleading to me. "Statistically self-similar" is practically an immediate consequence of most definitions of fractal, so "all fractals are self-similar" almost has a ring of tautology to it. Another reason I don't like "self-similar" as a definition is that Euclidean objects display self-similarity; this is a major flaw in using "self-similar" as a definition. I've put in a comment that "all fractals are self-similar in some sense", with a description of the different notions of self-similarity. I know we have to compromise in opening paragraphs, but defining a fractal as self-similar seems to be going just past the point of sacrificing accuracy. It could also be extremely confusing, when a different definition is given below in the article. Maybe trying to give a definition at all at first is bad. Perhaps it would be best to give a list of properties that most common examples have, and say, "a fractal generally means a geometric object having most, if not all, of the following properties", self-similarity would be one. This is done in the introduction of a textbook I have. Since there is no single definition of "fractal", maybe that's the best that can be done. Revolver


 * As we have opened up the issue of definition of fractals, I would like to comment on the sentence in the Definitions section that says Strictly, a fractal should have fractional (that is, noninteger) Hausdorff (or box-counting) dimension. Is this correct ? Doesn't the Peano curve has Hausdorff dimension 2, because it is a space-filling curve ? Also it has been proved that the boundary of the Mandelbrot set has dimension 2 (according to Mathworld). So there would appear to be fractals that have integer dimension. Gandalf61 09:44, Jul 13, 2004 (UTC)
 * Good point. I explain this in the article. The essence of the math definition usually taken is that the Haus dim > top dim, not that Haus dim is an integer. Revolver

Well, that's right, in strict mathematical usage (I put in the business about the Mandelbrot set, at one point). But I think everyone knows that the 'fractal' coinage was not about strict mathematical usage. Charles Matthews 10:25, 13 Jul 2004 (UTC)
 * Okay, that's fine, but saying "a fractal is a geometric shape that is self-similar" sounds a lot like a definition, so this is why I propose the list of descriptive characteristics in place of an attempt at a definition. Revolver 19:39, 13 Jul 2004 (UTC)

My main objection is not so much the use of "self-similar" to describe fractals, it is the use of only self-similar to do so. Given that "fractal" has no agreed fixed usage (or was never really intended to have one), then I think it's best to attempt an inclusive description rather than a simplified and misleading definition. In other words, if "fractal" has no precise mathematical usage, then this should be admitted up front, to everyone (not just math/physics people). As I said earlier, there is a list of about 4-5 properties of what are normally considered "fractals"; a particular given fractal may not satisfy ALL of these properties, but virtually all "fractals" satisfy MOST of them. Some fractals do satisfy all of them (the Cantor set satisfies just about any property you can think of, it's a good example to come up with properties). Some of these properties might include:


 * irregular, broken, or fractured (as opposed to smooth) geometric shape
 * infinitely fine detail (the more you magnify, the more detail you see)
 * some form of self-similarity (there are copies of the object at all scales)
 * recursive or iterative process generates the object
 * trying to measure standard length/area either gives 0 or infinite

and so on. All of these can be described in intuitive or relatively non-threatening language. And by saying that a fractal is something that satisfies most (but possibly not all) of these properties, it's still faithful to the way the term is used by math/physics people. It is a descriptive definition that takes longer than 2 lines or sentences, but I don't think it's asking too much. There are a lot of NON-mathematical objects that are like this (in fact, most), satisfying most of a list of properties, but usually not all. Most sets in real life are fuzzy. :) Revolver 22:16, 13 Jul 2004 (UTC)

Well, as I put it before, there is fractal the 'scientific concept', i.e. something that a chemist would be happy to discuss at a semi-formal level; and there is the actual mathematics of the point sets. These things have now diverged a little, you could say. Charles Matthews 07:35, 14 Jul 2004 (UTC)
 * I know that the definition Mandelbrot gave (Haus-Besic dim > top dim) is the most common or popular mathematical definition given, but is it really the only one offered? Mandelbrot wasn't satisfied with it himself. I guess what I'm asking is, "Is there really a standard definition people assume?", when someone writes "fractal" in a math paper, do they assume everyone knows what they mean, or do they say this explicitly? (no one has to clarify what a "group" or "top space" is, e.g.) I'm asking because I really don't know. (I've read textbooks on fractal geometry, but not any actual papers.)
 * As for the distinction between 'scientific concept' and 'math of point sets', I think the 'scientific concept' should dominate the opening paragraph. Not just because this is how most people use the term, but because the usual math def requires layers of other defs. Certainly, I think it was a good idea to remove mention of "Haus-Besic dim" and "top dim" from the opening paragraph.
 * If the informal, 'scientific concept' is the definition used in the opening paragraph, I still think even this non-mathematical usage should include more than just self-similarity. Certainly, when chemists are discussing "fractals" at a semi-formal level, they are thinking about irregular, fractured shapes, infinitely fine detail, and recursive or iterative processes, in addition to self-similarity. Maybe not as much, but this is part of their mental toolkit. I just feel that including a handful of other properties like this gives a richer description of the 'scientific concept'. Of course, it can be mentioned then that in mathematics, there are more precise definitions, which can be addressed later in the article. Revolver 08:44, 14 Jul 2004 (UTC)


 * So how would folks feel if the sentence about having fractional Hausdorff dimension were removed from the Definitions section, and replaced by saying that "Hausdorff dimension > tolopological dimension" is one possible mathematical definition ? I feel that the whole "fractional dimension" issue is a red-herring - it arbitrarily excludes objects that satisfy all other definitions of "fractal", and it seems to be based solely on the misapprehension that "fractal" is a contraction of "fractional dimension". Gandalf61 09:30, Jul 14, 2004 (UTC)
 * I agree completely. The usual def, the one Mandelbrot originally gave, was the Haus dim > top dim, not "noninteger Haus dim". It just happens that most of the time, when the inequality is strict, the Haus dim is not an integer...most of the time. That sneaky Mandelbrot. Revolver 09:57, 14 Jul 2004 (UTC)

Well, said 'misapprehension' seems to be been carefully designed in by Mandelbrot. Anyway the article should go over all this, to deserve 'featured article' status on grounds of completeness of coverage; while keeping the intro paragraph non-scary. Charles Matthews 09:47, 14 Jul 2004 (UTC)


 * I have changed the Definitions section so that it lists three possible definitions (self-sim, non-integer dimension, Hausdorff dimension > tolopological dimension) along with the shortcomings of each one. Re-reading some of Mandelbrot's papers, I have noticed that he describes the Hausdorff dimension as a fractional dimension - I take this to mean that it is a measure of dimension that may take non-integer values (as opposed to topological dimension, which is always an integer). I can't find any evidence that Mandelbrot intended to exclude objects with integer Hausdorff dimension from his definition of fractal, as long as they satisfied other conditions for being a fractal. Gandalf61 09:34, Jul 15, 2004 (UTC)

Intro reads like a Powerpoint slide
The list of points is terrible writing. There must be a better way to do this. Intros that read like Powerpoint slides are the opposite of good writing. I realise it was turned into this by the death of a thousand tweaks, but there's gotta be a better way that's both accurate and readable - David Gerard 21:19, 4 Aug 2004 (UTC)
 * Personally, I don't see anything wrong with it. Actually, my attitude is the opposite of yours --- I generally think writing in point form is more readable than paragraphs.  In this case the 5 points offer a clear, convenient checklist for deciding whether a given object is or is not a fractal. --Shibboleth 00:54, 5 Aug 2004 (UTC)
 * My opinion is that the current intro has reached a reasonable compromise between being both accurate and concise. But if you can re-write it without (a) losing content or (b) turning it into a mini-essay (which takes us back into the "can't we summarise this ?" loop) then go for it. Gandalf61 09:37, Aug 5, 2004 (UTC)

I've rewritten it. There is a fair chance I've missed something or interpreted something incorrectly, so please edit the lead further. The list was generally inappropriate, all the items were really the same thing. The the nature of fractals really only depend on one property (just like they only depend on one equation); their recursiveness. [[User:Sverdrup|❝Sverdrup❞ ]] 14:15, 16 Aug 2004 (UTC)
 * I was the one who basically came up with the "Power-point" presentation. I don't think it's bad writing style, where is it written (in Elements of Style??) that bullet points are bad style, if used judiciously? I was trying to reach a compromise, as Shibboleth mentioned, between two extremes &mdash; on the one hand, an intro that would please a finicky mathematician but be totally incomprehensible to most people, and on the other, an intro that would be completely understandable and pleasant to everyone, but be superficial and contain subtle fundamental inaccuracies. You are quite wrong to say that everything on the list is the same. My point is that "fractal" really is a fuzzy term, even in math it's just a catch-all for a family of definitions. The key part for me was, "satisfy most, if not all". I understand and confess that I approach this from a technical bias just because of my training, but I also thought it was important to make the point early on that it's a fuzzy term. Is something a fractal? Well, check the list. If it seems to have most of those, then probably yes. If it's missing one, so what. If it only satisfies ONE or TWO, it's probably not. But, to just say, "A fractal is a recursively defined object of infinite detail and self-similarity, etc., etc." may give people the impression that if something isn't recursively defined, or isn't self-similar, then it's not ever considered to be a "fractal", and that just isn't true. Revolver 05:06, 13 Oct 2004 (UTC)
 * Actually, the entire introduction as is currently stands seems pretty good, at least with regard to accuracy. I might want to change or reword a couple things (how is something broken up in a "radical" way?), but it seems to be going in a good direction. Revolver 05:28, 13 Oct 2004 (UTC)

Well, no. There are recursive things that aren't at all fractal-like and fractal things that aren't at all recursive (for example, randomised constructions like Brownian paths). One day we could agree to have this article written by someone who actually knows something about it; but I suppose we have to try the other ways first. Charles Matthews 15:38, 16 Aug 2004 (UTC)

The caption for the picture of Koch snowflake is not clear
The caption for the picture of Koch snowflake is ambiguous, errorsome or can be simplified to the extent that people like me can also understand. -- Sundar 04:19, Aug 19, 2004 (UTC)
 * Better now? [[User:Sverdrup|❝Sverdrup❞ ]] 04:41, 19 Aug 2004 (UTC)
 * Much better. Thanks, Sverdrup. -- Sundar 04:49, Aug 19, 2004 (UTC)

Not happy
Well, I'm not happy with the page, even though it is a featured article today. The 'definitions' business has gone off track, and the tone is too propagandist.

Charles Matthews 08:44, 19 Aug 2004 (UTC)

The coast of Britain ? Or is it Brittany ?
I'm pretty sure the original paper by Mandelbrot does not refer to the coast of Britain (in French : Grande-Bretagne), but the coast of Brittany, a region of France (in French : Bretagne).

Can someone confirm ?

--Stephan Leclercq 10:24, 19 Aug 2004 (UTC)


 * Sorry, but the title of Mandelbrot's 1967 paper is definitely How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. It was first published in the journal Science in English, so this is its original title, not a translation. The research on the lengths of coastlines to which Mandelbrot refers in this paper was carried out by Lewis Fry Richardson. Richardson measured the lengths of various coastlines and "natural" borders, including the West coast of Britain, which he defines as being from Land's End to Ducansby Head. But there is no reference to Brittany in either Richardson's research or Mandelbrot's paper. A Google search finds various references to Quelle est la longueur de la côte de la Bretagne?; if, as you say, la Bretagne means Brittany, then these must be the result of mis-translation in the opposite direction, from English to French. Gandalf61 09:04, Aug 20, 2004 (UTC)

"Measuring" the area/volume/... of a fractal
It may be just me, but I think the statement "Classical attempts to measure the size of a fractal's perimeter, area or volume fail, due to the lack of definite limit of detail." is untrue, or misleading at best. It's perfectly possible to determine the area of a Koch snowflake, for example, isn't it? -- Schnee 11:29, 19 Aug 2004 (UTC)
 * I agree that the statement is somewhat incorrect or misleading at least - you can determine the area of a snowflake, Britain, or any other line fractal. A two dimensional fractal you could determine the volume of but not its surface area.  Etc.  So depending on what kind of fractal you are dealing with, you may not be able to determine its perimeter, area, or volume. --Ignignot 18:03, Aug 19, 2004 (UTC)
 * I think you mean "a three-dimensional fractal you could determine the volume of", right? Anyhow, I think the surface area (that is, the n-1 dimensional surface in the general case) can be measured, too: it just usually will be infinite. I am not sure exactly what's meant by "classical" attempts, but I think taking a limit surely is not *that* modern an invention - or is it? -- Schnee 20:57, 19 Aug 2004 (UTC)
 * A stab at making a difference in classic and "fractal" ways: Coaslines cannot be compared, because both are infinite - this is a classical measurement. If we want to tell which is longer, we look at which has the largest fractal dimension, and we can call that one the longest. Or? [[User:Sverdrup|❝Sverdrup❞ ]] 22:00, 19 Aug 2004 (UTC)
 * Not necessarily, I think. Isn't the fractal dimension more like a measurement for the degree of convolutedness? In any case, the statement that you can't measure the size of a fractal's perimeter seems untrue to me. You can; it will usually (?) be infinite, but that's not an invalid measurement. -- Schnee 22:05, 19 Aug 2004 (UTC)
 * Did I miss this before, this "degree of convoludedness" the "fractal dimension" is what I'm referring to in the "dimension" subsection below. How is it calculated? I would like to know for myself, but should (could) it go into the page too?McKay 17:04, 20 Aug 2004 (UTC)
 * The fractal dimension is just the Hausdorff dimension. In fact, it's used to define fractals mathematically, too: a fractal is a set where the Hausdorff dimension strictly exceeds the topological dimension. -- Schnee 12:40, 21 Aug 2004 (UTC)

By "classical attempts to measure perimeter, area, volume fail" is meant the following. Take the Koch snowflake. I mean, take the boundary, not the interior. Of course we can calculate the area of the interior. But the boundary is different. If you try to find its "perimeter", you get infinity. If you try to find it's "area", you get zero. In any case, you haven't found a good measure of size -- classical measure, i.e. perimeter, area, volume. You can measure the size of the Koch snowflake, but not by classical geometry. Revolver 05:10, 13 Oct 2004 (UTC)
 * Yes, zero and infinity are "invalid" measurements. They are, from any practical viewpoint, useless. Revolver 05:11, 13 Oct 2004 (UTC)

Dimension
In Jr. High School, I remember calculating the dimension of a curve. Something about calculating the difference between distances when using different sized "rulers". Does this sound familiar to any of you deep math guys (I was going to just say "math guys", but I consider myself one of them, so I thought "deep math guys" was more appropriate)McKay 00:49, 20 Aug 2004 (UTC)


 * See Hausdorff dimension. David.Monniaux 21:09, 20 Aug 2004 (UTC)

Grammar check
From the article: That is, the traditional mathematical methods zoom in, in order to simplify the local picture; fractals flag the way this may fail when ever-finer detail is seen, with no definite limit.

This sentence seems wholly ungrammatical, and I don't know what it's supposed to be. --Golbez 17:02, 20 Aug 2004 (UTC)

Not totally, surely. No doubt it could be made less colloquial, and easier to parse. Charles Matthews 17:08, 20 Aug 2004 (UTC)

Copyright free images of fractals
I have got a load of images at my gallery page (in swedish but most of the titles are in english and also the titles of the pages has an english translation). Some of them are uniqe fractals invented or modified by me. All images at the pages are rendered from my own code and are free to use for anyone. If you find something useful then do not hesitate to make a copy and upload it to here. // Solkoll 07:04, 9 Sep 2004 (UTC)

Fractals/Ancient philosophy
I find the new section relating fractals to some ideas in ancient or modern philosophy a bit hard to swallow. I don't understand what the given examples have to do with either mathematical fractals or "fractals in nature" (coastlines, etc.) It's generous to say it's stretching. The link confirms my fears. The paper attempting to link Derrida and chaos theory is particularly distressing (and amusing.) Revolver 01:27, 15 Oct 2004 (UTC)


 * In chaos science we have a bevy of scientific words and phrases to come to grips with; fractals, phase space, bifurcation, intermittencies, periodicities, aperiodicities, Butterfly Effects, turbulence, linearity, non-linearity, attractors, strange attractors, points, pendulums, complex systems and non-linear dynamics, to mention just a few. Chaos cuts away at the tenets of Euclidian geometry and Newton's physics. It is out to replace the fundamental traditional knowledge in physics and mathematics with the new kind of knowledge necessary to understand the elements of motion rather than matter, i.e., pendulums, fluids, electronic circuits, lasers. We can think of chaos science as investigating `becoming' rather than `being'.  -- Laurie McRobert, Derrida, Hegel, and Chaos Science
 * I rest my case. Revolver 01:35, 15 Oct 2004 (UTC)

In Response
[EDIT: Removed by poster]

...Second, as far as the link: I believe the Fractal Metaphysics Web Page represents a unique and valuable resource. Every science and every geometry has a sister philosophy, and I believe one of the most important legacies of chaos theory will be in expanded dimensions of human thought.

It's unfortunate that Revolver chose to focus on one of the less mathematical external links leading out from FMWP and to use it as a reason to deride the enterprise as a whole. I do believe that less mathematically rigorous applications of chaos theory can have validity. But for those who disagree, there are also articles on the site dealing, in a mathematically valid manner, with the construction of the Koch snowflake, the geometric structure of the Sierpinsky Gasket, and several different constructions of the Harter-Heighway dragon --as well as the relationship of each of those to (respectively) the nature of motivation, social structures, and the nature of reality itself.

In summary, I think that both contributions to the article were valid, and the use of one decontextualized secondary link to discredit the entire enterprise was a cheap shot.

kitoba Oct 19 5:27 EST


 * about the section. there is a farily lengthy article on recursion, which i think might be appropriate for an "ancient" example; the comparisons with plato and the biblical quotes to fractals are pretty facile. many fractal-ish structures exist throughout the world regardless of whether people mentioned them. Frencheigh 22:59, 19 Oct 2004 (UTC)

Kitoba, I actually think you have it backwards &mdash; fractals were rendered long before they were conceptualised. Anyone can sketch a few iterations of the Koch snowflake or Cantor set with pencil and paper (it's possible to do so without knowing anything about fractals). Moreover, computer renderings of these objects will not help you visualise them much better than pencil and paper renderings. It is true that visually extravagent examples requiring lots of computation (like the Mandelbrot set) were for the most part out of reach of human perception until very recently. But it was precisely the concept of recursion that allowed the rendering of fractals before the age of computers. As for the conceptual basis of fractal geometry, this is an extremely recent phenomenon, not developed until the 20th century.

'''I'll conceed this point. I think the observations were valid, but probably too tangential - K'''


 * Every science and every geometry has a sister philosophy, and I believe one of the most important legacies of chaos theory will be in expanded dimensions of human thought.

I'm not sure what exactly this is supposed to mean, or what "sister philosophy" is. Of course, I think the legacies of chaos theory will be in expanding human thought, but then again, I think the same way about modular forms, manifolds, and non-standard analysis, but of course, none of these are popular fads. "Chaos theory" and "fractals" seem to be attractors (no pun intended) in the public imagination for everything thought to be revolutionary about math, and this is why it's been given this strange aura and significance by postmodernists, poststructuralists, and literary theorists.

As for the other articles on the page, yes, I did look at them, and yes, I did choose the worst one. The most harmless was the one written by the math professor...while I'm not sure what the point was, the references to math were at least accurate. (Although the analogy between projections onto a subspace in Hilbert space and shadows on a cave seemed stretching.) Also, the constructions of the fractals you mention seem to be done fine &mdash; I just don't follow the grand metaphysical leaps in those papers. This is probably a fundamental disagreement between mathematicians and philosophers who choose to "apply" math to their work. The whole point in applying math is to apply it in a rigorous way. Mathematical ideas are preicse &mdash; if you lose the precision in mathematical ideas, you've lost the whole idea itself.

Upshot: I just don't see how this relates either to the mathematical basis of fractals or its application, and (as a mathematician) I find the co-option of the terminology of fractals and chaos by recent philosophers (in an explicitly mathematical sense) as an abuse that often demonstrates little understanding of the math. Revolver 00:16, 20 Oct 2004 (UTC)

Certainly the existence of the (ab)use of fractal geometry and chaos theory among philosophers is a true and verifiable fact &mdash; a short search among philosophy and literary theory journals will return a large number of results. Perhaps a mention of this, along with a mention of the ongoing antagonism/debate over this use (which is also a true and verifiable fact) along with a couple links to both said philosophical papers and criticisms. An extended discussion should probably go onto another page in any case, this is primarily about the math/science...and even then, longer excursions into the technical math discussion have been criticised, so this isn't aimed just at philosophy. Revolver 00:28, 20 Oct 2004 (UTC)

I think this is the crux of the matter. As with many unfamiliar conceptual leaps, it may be hard for someone to "follow" the connection between a metaphysical idea and a geometric one, but that does not mean that the connection is imaginary or invalid. Nor does the fact that some people have abused mathematical concepts to fit a metaphysical context mean that all such use is therefore abuse.

This article is about fractals, and as you yourself have pointed out, "fractal" as a concept has an impact that goes beyond the strictly mathematical.

However, I agree, such extensions of the idea of "fractal" belong in their own section, and some mention of their controversial nature should be made. kitoba Oct 20,2004 00:29 EST

EDIT: I've taken your advice and added a new page called fractal metaphysics.

Most of the above discussion seems to be semantics to me. These "ancient fractals" are actually fractal-like. Hyacinth 20:17, 20 Oct 2004 (UTC)