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Fractals 101, Part 1

By jd in Science
Wed Jul 20, 2005 at 11:36:19 AM EST
Tags: Science (all tags)

Fractals are well-known in rave art, popular culture and even the occasional movie. But what are they, where do they come from and what are we going to do with them now they are here?

In order to cover fractals in any meaningful way, I am going to first examine what lies behind them. The first, and most important, of these elements is "Chaos Theory", a branch of mathematics that is rather better known than it is understood.

Chaos Theory covers a class of mathematical systems that don't follow the usual rules. (Anarchistic equations!) Normally, mathematical functions are relatively well-behaved. You can find the gradient at a given point, for example. If you change the values you start with (your initial conditions) by just a little, you will change the values you end up with by just a little and in a predictable way.

Chaotic systems, on the other hand, throw the rules out the window, without opening the window first. For small enough initial conditions, they behave themselves and generally go to some steady state value. As you increase the values, though, something happens. The system will start to oscillate with period 2. Keep increasing the value, and it stays at that frequency, until you cross a specific threshold. Then the frequency doubles. It keeps on doing this, with the intervals getting shorter each time.

Translated into English, this means that if you take some function and then use the results as the new inputs to the same function, it will first bounce between two values. Then, as you increase the initial conditions, it will bounce between four possible values. Then eight, sixteen, thirty-two and so on. It will always be a power of two and it will always be in that order.

At this point, a careful observer—call them Mitchell Feigenbaum for the sake of argument—may notice that the ratio between the thresholds is fixed for that function. In other words, if your start point is P0, you first double frequency at P1 you double again at P2, and double a third time at P3, then (P1-P0)/(P2-P1) = (P2-P1)/(P3-P2), and so on for all of the times the frequency doubles.

A really good observer will then calculate this constant. It will be about 4.669211660910299067185320382047. Later, you might try different chaotic functions. You will get exactly the same behaviour and exactly the same ratio. The Feigenbaum's universal constant applies to ALL chaotic systems, in that interval where oscillation occurs. There are no exceptions and there are even fewer useful explanations.

OK, so what happens when we go beyond oscillation? Then we enter the realm of pure chaos. The system will be totally deterministic (it is pure mathematics, after all, and there are no random elements to it) but it is completely unpredictable. It will never repeat, it will never settle down, it will never do anything you might expect. If the initial conditions change at all, no matter how slightly, the chaos completely changes. It is almost as if you are looking at two completely different systems, not the same system with a very minor tweak.

This kind of result is called "sensitivity to initial conditions", though popular magazines will usually call it by the name of "The Butterfly Effect". It is not the end of the problem, though.

As you look closer and closer at the system, you will notice that the system isn't smooth. It is as seemingly complex on a fine scale as on a large scale. This means that there is no meaningful "gradient" to the function. Not just at one or two points, as happens with discontinuous functions in "normal" maths, but at ALL points in the system. The technical term for this is a Non-Differentiable Function.

But what causes all of this strange behaviour? Surely, there must be a simple explanation, some quirk in the maths. Well, the answer to that is yes, there is. The quirk is called a Strange Attractor.

Simply put, Strange Attractors are anomalous regions in the chaos that seem to pull the chaotic system in their direction. In some ways, this can be likened to gravity, but the effect doesn't always fall off with distance and is not always easily predictable.

There are Strange Attractors in normal mathematics too; we just don't really see them that way, as they are much more benign. Picture the numbers from zero to infinity. Pick one, any one, it can be a fraction or an irrational number, if you like. Now square it, and keep squaring the result. Numbers above 0 and less than 1 will fall towards 0. Numbers below infinity and above 1 will fall towards infinity.

Zero and infinity, then, are attractors for the square function. They will pull numbers towards them, and a stable boundary (1) exists where the pull is equal in both directions.

In chaotic systems, Strange Attractors might appear anywhere. Systems can orbit them, seemingly stable, only to be flung away the next moment. Other times, a chaotic system may stray too close and get pulled into these mathematical black holes. The boundaries of their influence are ill-defined and can seem to intrude into areas controlled by other Strange Attractors.

How does this lead us to fractals—the pictures people see on computers and t-shirts? Well, let's take this one step at a time. The Mandelbrot Set—the best-known fractal of all—is based on the chaotic system Z' = Z^2 + C, where Z and C are complex numbers, and Z' is the value of Z for the next time the calculation is made. The usual way to plot this is to plot the real component of Z as the X coordinate and the imaginary component as the Y coordinate. Then just sit back and watch how the point moves through the system.

There are lots and lots of Strange Attractors hiding in this seemingly trivial equation, and we shall see this in a moment.

If we plot how Z changes, every time we cycle round the equation, we will see one of a number of possibilities:

  1. It may fall into a Strange Attractor and stop.
  2. It may orbit one or more Strange Attractors indefinitely, never breaking free of them.
  3. It may do all kinds of loops and spins around a whole load of Strange Attractors, before escaping to infinity (which is just a regular attractor, not really a Strange Attractor).

When displaying a "classical" Mandelbrot picture, what you are doing is plotting how long the system takes to escape—if it does. Points that do NOT escape (the central plateau of the Mandelbrot Set, for example) are given some otherwise unused value to mark that they did not escape.

Points far from the center tend to escape very easily and quickly, which is why you get fairly boring, bland rings on the outside. Even there, though, you'll notice that as you approach the center, the pull from the Strange Attractors is enough to slow down the escape.

Once we reach the "interesting" part of the set, things go crazy. The function gets pulled in all kinds of directions, with absolutely zero regard for how things were even an infinitesimal distance either side. This is where you get all of the pinwheels, spirals and other strange effects. As we approach the edge of the plateau itself, the chaos worsens as the number and density of Strange Attractors grows.

Eventually, on the plateau itself, nothing can escape. Not even light. Oops, wrong topic. Seriously, within the region of the plateau, ALL starting points will be captured by one or more Strange Attractors. Nothing escapes from this region. If you go back to plotting Z'=Z^2+C, and throw something into that region, you will see that it never comes back out. It really is a mathematical Black Hole, but not due to some gravitational singularity, but rather to a large enough number of powerful enough attractors that nothing escapes.

One of the consequences of having these Strange Attractors pull the system in different directions is that the motion of the system is disjoint. Not just in a few cases, but essentially everywhere. This means that the fractal you produce is also non-differentiable—there is no gradient you can calculate at any point.

But there is another, more curious aspect. Something called "Self Similarity". Self Similarity means that if you zoom into a part of the whole, you will see something that resembles the whole. It won't be exactly the same, but it will be close.

Self Similarity is one thing that makes Fractals so curious. If the system is being thrown around by all these Strange Attractors - which can be anywhere - then how can a small fragment look like the whole? That would imply that Strange Attractors can't be randomly distributed, their distribution and the interaction between them must ALSO be self-similar. They drive the shape, so what the shape does, they must do also.

Of course, things get worse. At no point in a fractal does the pattern repeat itself, so although the Strange Attractors must approximate their positions on different scales, they cannot have exactly the same layout. Nothing repeats in a fractal. Ever.

However, remember that bit about sensitivity to initial conditions? The layout must not only change, it must change in a manner that produces similar results. Normally, with the system being so sensitive and all, any change at all would produce something totally different. Thus, not only must there be change, there must be change that largely (but not entirely) eliminates this sensitivity aspect.

That should be enough to curdle most people's brains at this time. Next story will be on self-similarity in the marketplace and why the free market is really a chaotic system. (This was first shown by Benoit Mandelbrot and now forms a rather obscure but fascinating branch of economic theory.)

The series will wrap up with a talk on using fractals to compress naturally-occurring data, especially images. (Briefly, this boils down to trying to produce a fractal that is very close to what you want and optionally storing differences between the fractal and the original. The fractal and the differences will require much less storage than the original data.)


Voxel dot net
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Related Links
o Fractals
o "Chaos Theory"
o Feigenbaum 's universal constant
o "sensitivi ty to initial conditions"
o Non-Differ entiable Function
o Strange Attractor
o "Self Similarity"
o Also by jd

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Fractals 101, Part 1 | 119 comments (81 topical, 38 editorial, 0 hidden)
Pick up a book.. (2.75 / 4) (#1)
by gyan on Tue Jul 19, 2005 at 02:45:25 AM EST

The Computational Beauty of Nature is pretty much the best introductory treatment of these topics.


Weird (none / 1) (#61)
by gavri on Wed Jul 20, 2005 at 12:37:24 PM EST

I never knew Amazon url's could be this short. Weird seeing one without a referral.

Blog Of A Socially Well Adjusted Human Being

[ Parent ]
Chaos and Fractals by Otto-Pietgen, et al (none / 1) (#85)
by ChadN on Wed Jul 20, 2005 at 11:14:17 PM EST

I just happen to finally be reading Chaos and Fractals, by Heinz-Otto Peitgen, Hartmut Jürgens, and Dietmar Saupe; still one of the best overviews of the field. I was set to have Otto-Peitgen as my Calculus 101C teacher at UC Santa Cruz, in 1989, which I was VERY excited about. I had another of his books on fractals, "The Beauty of Fractals". The Berlin wall came down, and he stayed in Germany for that term... Oh well. :)

[ Parent ]
How do you market fractal software? (3.00 / 2) (#27)
by Sesquipundalian on Tue Jul 19, 2005 at 05:38:09 PM EST

I'm just finishing up a piece of software that I wrote out of pure hobby interest. It's been under development for a few years now.

What started out as a Visual Studio project that I could use to play with John Conway's game of life has become what ammounts to "Office for Fractals" (I'm not sure what I'm actually going to call it yet).

In addition to standard grids based on squares, this framework also supports hexagonal, pentagonal and triangular grids, in both two and three dimentions. The package also features a well developed functional programming language and a built in JIT compiler (rather than creating your rule in pascal or c++ and having to compile a .dll, you create your fractals using a language that has been created exclusively for this purpose).

Other nice features include realtime audio processing (so your fractals can dance to WinAmp), scripted batch file processing (so you can use your fractals to morph folders full of pictures, or create new audio effects in wave files). You can easily create screen savers with cycling picture backdrops (your compiled fractal runs in a transparency layer, and it can actually see the picture). You can also create fascinating interactive Active desktop elements (little amoeba guys that crawl around your desktop and interact with the background).

The coolest thing I've done so far is create this sort of magic sandbox that sits on my desktop and morphs to music. You can draw on the fractal as it develops, or just let it sit there as an active desktop element that looks cool. A friend of mine apparently lost a whole weekend playing with this (this thing has a major lava lamp factor).

The JIT compiler supports libraries and I have already discovered a few hundred of my own original rules that other people can incorporate into their fractals. You can do all of the classics like Langton's ant, Brian's Brain, Forest fires and such, and feature-set wise it mops the floor with other packages like Mirek's Cellebration, or the Collidoscope.

So the question is, how the heck do you market something like this? I'd love to see the package take off like WinAmp or something. The screen savers are a breeze to create and even my non-technical SO has been making screen savers as presents for people. How do you sell this thing? Universities? Shareware? I've been scratching my head over this for some time now...

Did you know that gullible is not actually an english word?
Headshops. (3.00 / 4) (#29)
by Back Spaced on Tue Jul 19, 2005 at 06:15:00 PM EST


Bluto: My advice to you is to start drinking heavily.
Otter: Better listen to him, Flounder. He's pre-med.
[ Parent ]

Include a free hit of acid with every purchase. (3.00 / 5) (#32)
by Lanes Inexplicably Closed to Traffic on Tue Jul 19, 2005 at 07:07:39 PM EST

[ Parent ]
How does it compare to Fractint? (none / 1) (#37)
by localroger on Tue Jul 19, 2005 at 08:21:18 PM EST

Office for Fractals already exists, and it's freeware, and it's a labor of love on revision 20.0, so unless you have something head and shoulders above or different enough to spark a fad, it's hard to see how you would compete with their free as in beer sales model.

I am become Death, Destroyer of Worlds -- J. Robert Oppenheimer
[ Parent ]
I doubt it. (none / 1) (#38)
by Lanes Inexplicably Closed to Traffic on Tue Jul 19, 2005 at 09:19:24 PM EST

Many of the features he listed are not a part of FractInt. I suppose you could script them with some ugly Perl hacks on Unix, but his software package doesn't require anyone to do that.

[ Parent ]
Nope (none / 1) (#75)
by gyan on Wed Jul 20, 2005 at 07:37:40 PM EST

I think the most sophisticated fractal generator is Ultrafractal.


[ Parent ]
partial crippleware? (none / 1) (#42)
by vhold on Wed Jul 20, 2005 at 02:49:05 AM EST

Basically make a pretty much fully functional free version, but move some of the more professional pieces of functionality to the full registered version, and make some funky goofball features that people might imagine as being really cool.

Like uhmm..  

the scripted batch processing.

multipage printing to make giant fractal posters.

the winamp audio response mode has a little scrolling message every now and then (small enough to be not be totally annoying, but enough to look cheesy and lame in a club or bar)

[ Parent ]

And market it to bars as an ad display format? (none / 1) (#53)
by Sesquipundalian on Wed Jul 20, 2005 at 10:17:38 AM EST

That could work very well. I have some experience selling software packages to bars and clubs. Avertising-wise, you can actually do some pretty neat things with this (something with Brian's brain and "the army ants that march through vodka" droodle, perhaps). hmmmm some very good suggestions in this thread, thanks a lot and keep them coming!

I do have some higher end features that I was thinking of bundling into a "for pay" version. The cluster feature (where you join copies of the program together over a network and make huge fractals in a reasonable ammount of time), or the "symmetry draw" feature where you basically draw your own grid tiling.

Did you know that gullible is not actually an english word?
[ Parent ]
Bundling? (none / 1) (#45)
by forgotten on Wed Jul 20, 2005 at 03:40:55 AM EST

Get some company to include it with their software, or hardware. Eg, some company markets a media centre package, or even a simple dvd player.

I'm pretty sure thats how many people make their money. Both of the last dvd players I bought had nero bundled, as well as windvd.

Aside:windvd is shit. It never gets the aspect ratio correct. Why bother with crap like automatic imdb lookups for movies when the whole fucking picture is distorted.


[ Parent ]

Simple (3.00 / 3) (#99)
by X3nocide on Thu Jul 21, 2005 at 02:45:38 PM EST

Find a local printshop that can do posters, and is willing to negotiate a deal with you. Give the program away for free, and maybe spend some money advertising it. Allow people to share fractals, and include some code to send a particular step in a fractal to your printshop for a fee.

The keys to making this work are:

  1. A quality interface to making interesting fractals quickly.
  2. A way to keep users from taking those fractals to other printers.
  3. Purchasing a poster based on a fractal image should be as painless and simple as possible. If you're concerned about repeat sales, consider your shipping options.

[ Parent ]
huh (1.09 / 11) (#30)
by Tex Bigballs on Tue Jul 19, 2005 at 06:31:29 PM EST

So how's that accounting working out for you? (3.00 / 2) (#43)
by forgotten on Wed Jul 20, 2005 at 03:35:00 AM EST



[ Parent ]

He's not stupid (none / 1) (#74)
by JaxWeb on Wed Jul 20, 2005 at 07:25:46 PM EST

If he thought he understood, he was stupid. The article did not contain enough information for an intelligent person to feel informed.

[ Parent ]
I didnt mean that. (none / 1) (#76)
by forgotten on Wed Jul 20, 2005 at 08:26:31 PM EST

you are right, too. but i think it accidentally got posted before reediting. a lot of the comments clarify the text.


[ Parent ]

yes he is (1.00 / 5) (#110)
by n1ckn4m3 on Fri Jul 22, 2005 at 01:31:59 PM EST

his "response" to the article notwithstanding, he is by any objective measure an idiot.

[ Parent ]
Some Fractal/Chaos related applets (none / 1) (#31)
by nkyad on Tue Jul 19, 2005 at 06:56:22 PM EST

While Cynthia Lanius Fractal Lessons page lists some nice applets (the "Using Java" links in the menu to the left), they are quite simple and limited.

Aaron Davidson's Mandelbrot Set Java Applet is one of the best programs of its kind I've seen. Quite beautiful. And here is a list of many other Fractal/Chaos related applets.

Don't believe in anything you can't see, smell, touch or at the very least infer from a good particle accelerator run

Mandelbrot applet (none / 1) (#71)
by chroma on Wed Jul 20, 2005 at 06:50:02 PM EST

I wrote this a Mandelbrot browsing applet a few years back. It has some features not available in the applets you link to: you can zoom out in steps, manually enter coordinates, choose between fast and accurate rendering, and, best of all, you can enlarge the window to the whole screen.

[ Parent ]
Taylor Series Fractal (none / 1) (#82)
by cabin on Wed Jul 20, 2005 at 10:22:13 PM EST

Here is an applet (at the bottom) based on the Taylor Series approximation. It gives some very unique images.

[ Parent ]
Why I decided not to included it (aka bugreport) (none / 1) (#86)
by nkyad on Wed Jul 20, 2005 at 11:17:12 PM EST

I saw your applet, and it was indeed very good, probably the best interface and featurewise. Nevertheless it has a annoying bug, it won't close unless you close the browser (Win XP/Firefox). Do you know why?

Don't believe in anything you can't see, smell, touch or at the very least infer from a good particle accelerator run

[ Parent ]
Because (none / 1) (#87)
by chroma on Wed Jul 20, 2005 at 11:44:35 PM EST

I was too lazy to implement that. Or it might have had something to do with Java's security model; I don't quite remember.

Anyway, if you just navigate away from the original page in your web browser, the window should close. At least it does on Konqueror 3.3 on Linux, Mozilla 1.5 on Windows 2000, and IE 6 on Windows 2000.

Of course, running it just now (and I haven't really looked at it in a couple years), I found a few other bugs.

[ Parent ]

Tabbed browsing makes it worse, I guess (none / 1) (#88)
by nkyad on Thu Jul 21, 2005 at 12:05:06 AM EST

I just middle-clicked a link to your page and the applet poped up. It never occurred to me to go to that specific tab (of the 10 or 12 then open) and "navigate off". Nowadays I think its a 50/50 chance I will navigate off some page or open a new tab on an interesting link. Here in K5, for instance, given the interface, I usually navigate in one page while posting, moderating or reading articles, but I will open all external links in other tabs and all diary pages I care to read in separate tabs too.

And yeah, 1.1 (now that I actually read the page :)) had some wierd bugs, specially in applets, double specially in heavyly graphical applets. The recent 1.5 is far more stable and Java2D is making possible the rebirth of the applet as a content/app delivery medium (instead of the CVS-less Flash or Director/Shockwave).

Don't believe in anything you can't see, smell, touch or at the very least infer from a good particle accelerator run

[ Parent ]
Applets (none / 1) (#91)
by chroma on Thu Jul 21, 2005 at 01:02:16 AM EST

You're preaching to the converted. We just need a catchy new name for applets to compete with "AJAX", "Flex", and so on.

I considered doing a Swing/Java2D version of my applet, but I've got a bunch of other projects that interest me a lot more at this time.

[ Parent ]

Ugh, hardware... :) (none / 1) (#93)
by nkyad on Thu Jul 21, 2005 at 01:33:32 AM EST

Uncontrollable enviromental conditions, very untidy real physics (who wants real constant Gravity?), bolts, screwdrivers, wheels. Ugh. Hehe.

PS: Great projects (and pretty robots) there. Nice resume too - I mean, if we were on the same hemisphere I'd certainly consider hiring you :)

Don't believe in anything you can't see, smell, touch or at the very least infer from a good particle accelerator run

[ Parent ]
Hiring (none / 1) (#106)
by chroma on Thu Jul 21, 2005 at 09:21:06 PM EST

Outsource to the USA now while the dollar is weak!

[ Parent ]
Book recommendation (2.50 / 6) (#34)
by Lanes Inexplicably Closed to Traffic on Tue Jul 19, 2005 at 07:16:04 PM EST

I'm working through Nonlinear Dynamics and Chaos, by Steven Strogatz, right now as an aid to understanding the Three-Body Problem. Having already taken a course in differential equations, what I've read through so far is not terribly challanging to grasp, and the book is written without the expectation that the reader has taken a diff. eq. course, i.e. if you have calculus, you'll get along fine. It's heavy with applications of the various concepts it teaches, which means it will be much more useful and interesting to science and engineering types -- definitely worth checking out.

I agree (none / 1) (#102)
by a humble lich on Thu Jul 21, 2005 at 06:04:35 PM EST

This is by far the easiest to read math book I've seen. His section on Chaos is (in my opinion) a bit too short, but the book is an excellent introduction to general nonlinear dynamics.

[ Parent ]
great (2.75 / 4) (#50)
by fleece on Wed Jul 20, 2005 at 08:50:08 AM EST

My attention span is at it's shortest when i'm at K5 so if i can get through something from start to finish that's a good sign. I only have high school maths but you managed to explain it in a way that was interesting through to the end and actually made sense. good work!

I feel like some drunken crazed lunatic trying to outguess a cat ~ Louis Winthorpe III
great2 (3.00 / 2) (#67)
by calcroteaus on Wed Jul 20, 2005 at 02:27:30 PM EST

I agree...I don't come to kuro5hin (or any website for that matter) to become a physicist or mathmetician. While some of the people who are complaining are probably right that this article is inaccurate or not well explained, it provided me with a good understanding of the concept in layman's terms which has stimulated my interest.

[ Parent ]
If you think you understood it... (3.00 / 2) (#73)
by JaxWeb on Wed Jul 20, 2005 at 07:24:06 PM EST

There are three types of people. Those who read and realise the article does not say anything, those who read and think they understand [even though they cannot possibly understand because the article does not explain] and those who just plain don't understand.

[ Parent ]
You Forgot One... (none / 1) (#108)
by calcroteaus on Fri Jul 22, 2005 at 11:25:53 AM EST

Type Four...people who only provide critcism without positive input...how about posting your explanation of Chaos Theory which is interesting, engaging, and most importantly, accurate, so that we can all "understand" the subject better?

[ Parent ]
4.669211660910299067185320382047 (2.71 / 7) (#51)
by Viliam Bur on Wed Jul 20, 2005 at 09:09:53 AM EST

I completely fail to understand (from the article) what this number means, and how was it found. At best, you have skipped a very important part of explanation (or the whole explanation); at worst, you have just produced a piece of pseudoscience.

"Chaotic systems ... generally go to some steady state value. As you increase the values ... The system will start to oscillate with period 2. Keep increasing the value ... until you cross a specific threshold. Then the frequency doubles. ... (P1-P0)/(P2-P1) = (P2-P1)/(P3-P2) ... It will be about 4.669211660910299067185320382047."

It seems to me like you are describing an example, but the example is not included in the article. I am completely unable to discover what do you mean (if anyone can, please reply to this); cannot tell any example of the "chaotic system" that goes to stready value and doubles as 4.something... whatever this may mean.

I suspect this to be a pseudoscience, because this is how mathematical pseudoscience typically looks like: no examples, no equations, that suddenly one magic equation, one magic number (no obvious relation between the equation and number, just "trust me, it works, the Scientists say so!"), and a long text with a lot of buzzwords. Maybe some words and sentences are picked from the real mathematical texts, but this one does not make a sence.

A mathematical article without mathematics looks like something written by a person who does not understand the topic but likes the nice words and metaphors, addressed to the similar audience. So they can say: "Wow, I read an article about Strange Attractors... and they were really Strange, and Attractive, and I did not understand it, but I feel that I will write another article about the topic too, because it's so cool!" It looks cool, and makes people say: "Wow, it's great... it does not make much sense to me, but it contains a lot of nice words. Let's vote it up."

Please, don't. -1 nonsense

How it works (none / 1) (#55)
by Eight Star on Wed Jul 20, 2005 at 11:33:10 AM EST

Hit the Feigenbaums universal constant link, Then click on the Drawing the feigenbaum fractal link.

This page from Mathworld has a better picture of the whole fractal.

The equation is x=x^2-a , if you repeat it enough times, X will settle down into one or more values, depending on what A is. A is the input or 'initial conditions' X is the oscilator. The fractal image is a plot of the X values(vertical axis) that occur For different values of A.
4.6692.. is a ratio of the horizontal distance between two places where the graph spilts, and the distance to the next split. It does seem to be a significan number, the author is not a crank on this one.
If you can program at all, I recommend writing up a program to draw the fractal, and play with it a bit (what happen if you start the iteration at 0, and only draw the 5th point each time? 25th point?) You will learn more from that than this article contains.

[ Parent ]
totally missing context (none / 1) (#64)
by PigleT on Wed Jul 20, 2005 at 01:47:28 PM EST

Yeah, what's missing is the whole context of what *kind* of fractal is being described - iterated systems (IFS), or repeated-geometrical-transformation or whatever.

What's really missing is how an object that can be drawn using one system can also be drawn by the others (cf ferns or Sierpinski gasket/triangle).
~Tim -- We stood in the moonlight and the river flowed
[ Parent ]

The point (none / 1) (#103)
by a humble lich on Thu Jul 21, 2005 at 06:17:44 PM EST

The exciting part about Feigenbaum's constant isn't that the map x'=x^2+C has this scaling property. Rather, any one dimensional unimodal (it only has one hump) map has the same scaling close to the transition point. The maps x'=x^4+C or x'=sin(x)+C (0<x<pi) should scale the same way. <p> That is why it is interesting, because it describes unexpected universal behavior among all systems in its class. Now one is free to ask how useful it is since few physical systems are described by one dimensional dynamics, but that is another point.

Now only would I not call it pseudo-science, it isn't really even science, just pure math.

[ Parent ]

Grumble (3.00 / 14) (#52)
by manobes on Wed Jul 20, 2005 at 09:40:39 AM EST

I'm not a pure mathmatician, and not a fractal expert, but a few things about this caught my eye.

Chaos Theory covers a class of mathematical systems that don't follow the usual rules. (Anarchistic equations!)

That's certainly not true. They follow "the usual rules" just fine. They're not particulary suited to numerical analysis, is the trouble.

Normally, mathematical functions are relatively well-behaved.

Well, no, not really. Perhaps what you meant is "the functions you see in high-school/freshman year, are well behaved?

For example the function exp(-1/x) is not "well-behaved". There is an essential singularity at x=0. Yet it's a normal mathematical function.

If you change the values you start with (your initial conditions) by just a little, you will change the values you end up with by just a little and in a predictable way.

This is the critirea for a *linear* system, not a "normal" one (whatever that means).

For small enough initial conditions, they behave themselves and generally go to some steady state value.

Small relative to what?

The Feigenbaum's universal constant applies to ALL chaotic systems, in that interval where oscillation occurs. There are no exceptions and there are even fewer useful explanations.

"all"? Are you sure? I don't know, but from your description, and the linked page, I would guess it only applies to systems with one degree of freedom. Like I said, I'm not an expert, but I certainly can't understand how one would apply what you said to a system with (say) 4 different oscillators, each with their own frequency.

No one can defend creationism against the overwhelming scientific evidence of creationism. -- Big Sexxy Joe

I have to agree (2.66 / 3) (#58)
by Coryoth on Wed Jul 20, 2005 at 12:12:20 PM EST

I am a pure mathematician, but not a fractal expert, and I have to agree with all your points.

While the article provies a nice introduction I wouldn't recommend it for more than just giving you the rough flavour of the concepts invloved.


[ Parent ]

It's important to note (3.00 / 4) (#84)
by jd on Wed Jul 20, 2005 at 10:25:08 PM EST

That K5 has more Joe Average's than it has pure mathematicians with a PhD in chaos theory. As such, the article was written with the audience in mind, rather than being an absolute purist treatsie on the subject.

It's also important to note that I labelled it a 101. In other words, it is a class for beginners, as an introduction to a subject. I cannot think of any school where a 101 is "correct" in any rigorous sense, and even many degree-level courses are wildly inaccurate when you get into the nitty-gritty.

My objective with the Fractals 101 is to bring together terms that are reasonably well-known but not generally explained or connected. If the course as a whole is as well-received as this specific article, then I will do a 201 course, which will be more "precise" but which will definitely require people to have heard more than just the names before.

There are a lot of "WHAT" questions answered by the article, but very few WHYs or HOWs, and next to nothing on the exceptions or the peculiarities.

An example - for those familar with Julia sets, you will no doubt be familiar with regions which look odd. Round areas that are divided into 8 different sections, where the sections alternate in value. What is going on in those regions? How can those exist, when the rest is apparently total disorder?

Another example - I mentioned that Fractals and Chaotic Systems don't repeat. They are not cyclic functions. How does this jive with the Sierpinski series of functions? Or with L-systems, or IFS systems, or anything else of that kind?

How do you actually go about calculating a fractal dimension for something? Especially in the Real World, where systems do NOT go on forever?

When you start getting beyond the James Gleik levels of abstraction into the really heavy-duty stuff, you're talking about something generally covered in the final year of a BSc degree program. It can be done on K5, and if the series proves popular enough, I'll prove that by doing that.

But to START there? When there are plenty of K5'ers who still confuse fractals with religion? Uh, no. You start with a foundation course, a very light introduction, which you can then work from.

[ Parent ]

Thank you (3.00 / 2) (#92)
by LodeRunner on Thu Jul 21, 2005 at 01:32:20 AM EST

I really enjoyed the light style and I'm looking forward to the next installments.

"dude, you can't even spell your own name" -- Lode Runner
[ Parent ]

Your major failing with this article (3.00 / 3) (#107)
by Lisa Dawn on Fri Jul 22, 2005 at 04:23:46 AM EST

... is that you make too many assertions, and too many of them are incorrect.

Even at the "James Gleik" level, you've made some really terrible mistakes. You don't need to explain everything, just keep your simplifications correct.

[ Parent ]

"All chaotic systems" (3.00 / 6) (#104)
by a humble lich on Thu Jul 21, 2005 at 06:25:17 PM EST

This is definatly not true. Feigenbaum's scaling in only true for one dimensional unimodal maps. While it is remarkable that it is universal in that group (and that all member's of the family scale with the same number of 4.blah), most physically relavent systems are not one dimensional.

On the other hand, even in larger systems the period doubling bifurcation route to chaos is quite common, but with a different scaling between doublings.

[ Parent ]

what what what? (2.00 / 5) (#54)
by So Very Tired on Wed Jul 20, 2005 at 10:22:20 AM EST

At this point, a careful observer--call them Mitchell Feigenbaum for the sake of argument--may notice that the ratio between the thresholds is fixed for that function. In other words, if your start point is P0, you first double frequency at P1 you double again at P2, and double a third time at P3, then (P1-P0)/(P2-P1) = (P2-P1)/(P3-P2), and so on for all of the times the frequency doubles.

God damn it you mathematicians make me sick. At first glance, your writings seem to make sense. On on further analysis, it's obvious that the only people that could understand this garbage are your fellow mathematicians."Fixed for that function?" "Start point is P0" (START POINT OF WHAT??) etc

I'll try to explain (none / 1) (#57)
by Eight Star on Wed Jul 20, 2005 at 12:09:10 PM EST

See my Other Comment Fixed for this function means that there is a structure to the graph. The ratio of the distances between consecutive splits is constant. (and in fact is constant for other similar functions as well.) The Start point, when I drew it, was 0, at the far left of the graph. (look at the mathworld graph in my other post) P1 would the the value of A (the input variable, Mu in the mathworld graph) where X (the oscillator) first starts to oscillate. (between P1 and 0, X would settle down to one value) P2 and P3 are subsequent values of A where the frequency doubles.

[ Parent ]
Re:what what what? (none / 1) (#62)
by scliffster on Wed Jul 20, 2005 at 12:39:23 PM EST

I don't think that this was written by a mathematician, or at least not a mathematician that was lucid at the time of writing:)

[ Parent ]
Happens Anywhere (none / 1) (#65)
by hardburn on Wed Jul 20, 2005 at 01:58:46 PM EST

Groups of humans tend to create jargon which provides very efficient communication within the group, but is totally incomprehensible to an outsider. This is hardly unique to mathematicians. Deal with it.

while($story = K5::Story->new()) { $story->vote(-1) if($story->section() == $POLITICS); }

[ Parent ]
Yeah, but (3.00 / 2) (#66)
by So Very Tired on Wed Jul 20, 2005 at 02:01:24 PM EST

Writers have their own jargon - phrases and words like context and "writing for your audience" . I'm looking around the audience tonight and I'm just.not.seeing.the.mathematicians.

[ Parent ]
Interesting topic, horrible writeup -1 (2.00 / 3) (#56)
by Fon2d2 on Wed Jul 20, 2005 at 11:52:11 AM EST

Or I would've voted -1 had I not been too late.

For those wondering about the 4.something magic number and all that gobbeldy-gook, it is not pseudo-science. It is just very horribly explained.

I believe he's talking about bifurcations and the logistic function. I'm currently trying to refresh my memory on this topic so an explanation will not be forthcoming from me at the moment. These two webpages, while not the easiest to understand, do seem to actually explain what he is describing.

Shameless Plug (none / 1) (#60)
by Chairman Kaga on Wed Jul 20, 2005 at 12:31:05 PM EST

My fractal website:
Fractal Recursions
I also have animations there.

Shameless fractal background pimpage! (none / 1) (#63)
by rianjs on Wed Jul 20, 2005 at 01:12:39 PM EST

A friend of mine from a forum made these and I've been hosting them. Lots of nonstandard resolutions for you monitor freaks.

  • 1024×768
  • 1280×1024
  • 1600×1200
  • 1680×1050
  • 1920×1200
  • 2560×1024
  • 3200×1200

No 10x16 Love? (none / 1) (#100)
by adavies42 on Thu Jul 21, 2005 at 05:40:53 PM EST

Very nice, but I note there's no renders in Mac laptop resolutions, like 1280x854. I'll use the 1280x1024 ones, but it would be nice to not have to squash/trim.

[ Parent ]
Apple must be on crack. (none / 1) (#109)
by Farting in Elevators on Fri Jul 22, 2005 at 12:06:36 PM EST

That is the most oddball resolution I've ever seen. It doesn't conform to a typical aspect ratio like 16:9; it doesn't even seem to be one of the lettered standards.

[ Parent ]
Check out the URL you gave (none / 1) (#114)
by LodeRunner on Sun Jul 24, 2005 at 02:09:36 AM EST

i386.info. You're listing PC video standards.

gasp I just rememebered Apple is switching to Intel. Nevermind.

Heck, it was so fun when Apples were different than Amigas were different than Ataris were different than IBM-PCs, etc... variety, choice... (yeah, I also remember incompatibility, but I honestly think it was a lesser evil than the blandness we're sunk into.)

"dude, you can't even spell your own name" -- Lode Runner
[ Parent ]

great2 (none / 1) (#68)
by calcroteaus on Wed Jul 20, 2005 at 02:30:00 PM EST

I don't come to kuro5hin (or any website for that matter) to become a physicist or mathmetician. While many of the people who are complaining are probably right that this article is inaccurate or not well explained, it provided me with a good understanding of the concept in layman's terms which has stimulated my interest.

Fractals and Chaos (3.00 / 2) (#69)
by Arkaein on Wed Jul 20, 2005 at 02:35:42 PM EST

I'm no expert on fractals, but the statement "Nothing repeats in a fractal. Ever." doesn't sound right to me. I've not studied the deeper mathematical aspects of fractals or Chaos Theory, my knowledge of fractals mostly comes from brief coverage in computer graphics classes. Some of the fractals I've learned about had nothing to do with chaos theory as far as I know.

One example is the Sierpinski Gasket. this type of fractal is generated in a much different manner than something like the Mandelbrot fractals. It is generated by essentially a recursive procedure, and in this example is not just self similar but use an identical pattern at each level of refinement. Another well known example is the fern branch fractal, where you simply take a gently curved line segment, pick a set of evenly distributed points and place smaller but otherwise identical curves sticking out to either side. Repeat this process for an arbitrary (theoretically infinite) number of steps.

Another example is terrain generation. Fractal terrain can be generated by starting with a low resolution noise-filled height map, and modifying this map by layering a series of successively higher resolution height maps on top of it. Each height map can use the game generator function, only the magnitude is different. The result is a fairly realistic appearance of hilly or mountainous terrain.

The generators for fractal terrain use (pseudo-)random variable, and so I believe that the concept of attractors cannot apply to this type of fractals. As far as strictly procedural fractals, attractors may exist here but not be necessary to understand merely how to produce the fractal. In any case I believe the exact nature of a fractal's self similarity depends on the type of fractal, and I'm not sure if all types of fractals mesh with Chaos Theory as in the article. I hope the author or someone else with more knowledge on the subjects can expand on this and correct any possible misconceptions.

The ultimate plays for Madden 2005

Pseudo-Fractals (none / 1) (#116)
by PhilHibbs on Tue Jul 26, 2005 at 11:37:27 AM EST

The generators for fractal terrain use (pseudo-)random variable, and so I believe that the concept of attractors cannot apply to this type of fractals.
In which case it isn't a true fractal, it's a faked approximation of a fractal.

[ Parent ]
Evidence? (none / 1) (#119)
by Arkaein on Fri Jul 29, 2005 at 10:20:05 PM EST

In which case it isn't a true fractal, it's a faked approximation of a fractal.

Got any evidence to back this up? I thought you might be right and did some searches on the matter, but could not find a single statement that indicates that fractal terrain (such as Brownian motion generated terrain) are not true fractals, though it seems logical if the definition of a fractal requires a strange attractor (as my searches did indicate that the Sierpinski Gasket has an attractor). However, the following definition from the sci.fractals FAQ does not mention an attractor as being a requirement for fractal classification:
A fractal is 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. Fractals are generally self-similar
and independent of scale.

This definition would seem to indicate that fractal terrain is a fractal, approximated only in terms of level of detail as all computer generated fractals must be.

The ultimate plays for Madden 2005
[ Parent ]

Run xaos! (none / 1) (#70)
by Peaker on Wed Jul 20, 2005 at 03:56:41 PM EST

Its a really cool fractals generator. It also has an impressive "tutorial" which is a great presentation introducing Fractals.

Fractional dimension ? (3.00 / 2) (#72)
by bugmaster on Wed Jul 20, 2005 at 07:14:20 PM EST

I thought that fractlas were called "fractals" because they have fractional dimensions ? The Sierpinsky triangle and that snowflake thing are fractals, too (in fact, they're the O.G. fractals, so to speak), and they are a lot simpler than the Mandelbrot set and other complex functions.
Yes and no (3.00 / 2) (#77)
by forgotten on Wed Jul 20, 2005 at 08:30:03 PM EST

most do, but the mandlebrot set has integer dimension. Its boundary has dimension 2, instead of the 1 point something, because it is "very" fractal.


[ Parent ]

If ya wouldn't mind a question (none / 1) (#78)
by SageGaspar on Wed Jul 20, 2005 at 09:33:31 PM EST

I'm an undergrad math major, and if you don't mind me asking, do you know an actual rigorous mathematical definition of fractal?

Mathworld lists it as having some property of self-similarity but never goes on to define self-similar. I'm thinking maybe there's always some sufficiently small proper subset homeomorphic to the entire set under the metric subspace topology? Or maybe you can perform rigid motions of the plane and dilations to get the entire set within some small epsilon of some sufficiently small proper subset under the Hausdorff metric?

On a side note, I took an undergrad discrete dynamics class from one of the leading guys in the field, and he hated talking about chaos, partially because so little can be said about a chaotic system, partially because of the way it haphazardly gets thrown around. And even then, there are multiple definitons of what exactly it means for a system to be chaotic.

[ Parent ]
its like pornography. (3.00 / 2) (#80)
by forgotten on Wed Jul 20, 2005 at 10:02:47 PM EST

you know it when you see it.

I don't think there is a strict definition. i dont think your suggestion would work for anything except a cantor-type set construction. Plus i am not sure if it could be consistent with topological transitivity, which is a characteristic of many systems one would like to describe as chaotic.

I heard much the same thing when I was taught nonlinear dynamics. he held up a copy of the paper "period 3 implies chaos" and warned that it was a field where papers needed to have catchy titles to get noticed.


[ Parent ]

The rigorous mathematical definition (3.00 / 3) (#81)
by jd on Wed Jul 20, 2005 at 10:03:05 PM EST

The formal definition is actually amazingly simple. The Hausdorff Dimension (ie: the fractal dimension) must be strictly greater than the Euclidian Dimension.

The non-rigorous, non-formal but actually useful definition is that there are two or more Strange Attractors in the system, such that anything within the vicinity of one Strange Attractor will be affected by the other.

(An example of a two-Attractor system would be the Lorenz Equations.)

This definition is useful, because you can look for Strange Attractors in both physical, real-world systems and mathematical ones, whereas the Hausdoff Dimension is extremely hard to compute for anything other than a mathematical system.

[ Parent ]

Sierpinsky ? (3.00 / 2) (#89)
by bugmaster on Thu Jul 21, 2005 at 12:56:22 AM EST

The non-rigorous, non-formal but actually useful definition is that there are two or more Strange Attractors in the system, such that anything within the vicinity of one Strange Attractor will be affected by the other.
Where are the strange attractors in Sierpinsky's triangle, and that snowflake thing ? If they're there, how would I find them, mathematically speaking ?
[ Parent ]
Partial explanation? (none / 1) (#96)
by SageGaspar on Thu Jul 21, 2005 at 10:23:48 AM EST

I'm sure you'll get a better answer from someone more knowledgable than me later, but it seems like the perspective of strange attractors would assume that you have some sort of function that generates the fractal.

If I just describe a subset of the plane with Hausdorff dimension > Euclidean dimension, then I'll also have a fractal, which is why that's the rigorous definition. I'm not sure that there's some canonical map associated with a fractal or you can always come up with a map that generates the fractal (although that would be neat).

For example, I can generate the Cantor set (or something very, very close to it) by looking at the points that leave the interval [0,1] under iterations of certain quadratic maps of the form y=a*x*(1-x). I forget which ones exactly, I think a > 4. But then I can also describe the Cantor set as a simple limiting process using subsets of the plane, cutting out the middle intervals, in which case there's no map at all (or at least no immediately obvious one).

In either case, I'm not sure what a strange attractor would look like in a one-dimensional space.

[ Parent ]
Strange Attractors (none / 1) (#101)
by a humble lich on Thu Jul 21, 2005 at 06:00:58 PM EST

The simple answer is no.

There is a distinction between fractals and strange attractors. I like the definition earlier of a fractal being any object with a hausdorf dimension greater than its euclidian dimension. A strange attractor on the other hand is something which is only seen in dynamical systems (solutions to differential equations or iterated maps). Most (possibly all, I'd need to think about a bit more) strange attractors are fractals, but not all fractals are strange attractors. The Cantor set and the Koch snowflake are just geometrical objects, and need no dynamical system to generate them.

Now there might be dynamical systems that could have attractors that are like the Kock snowflake are Cantor set, but that is not needed--the sets are defined outside of any dynamics.

[ Parent ]

OK (1.12 / 8) (#79)
by JESUS JONES on Wed Jul 20, 2005 at 09:56:15 PM EST


I love you /nt (1.50 / 2) (#83)
by uptownpimp on Wed Jul 20, 2005 at 10:25:07 PM EST

My name is actmodern and I approve of this message.
[ Parent ]
You sir, (none / 1) (#105)
by monkeymind on Thu Jul 21, 2005 at 08:29:51 PM EST

are no DUXUP.

That is all.

I believe in Karma. That means I can do bad things to people and assume the deserve it.
[ Parent ]

Fractint (2.66 / 6) (#90)
by bugmaster on Thu Jul 21, 2005 at 12:59:19 AM EST

The absolutely definitive fractal generator is Fractint. This is your non-stop, all-platform fractal shop. I can't believe no one's mentioned it yet... er... unless they did mention it, and I'm just blind.
YAFJA - Fractal Audio! (none / 1) (#94)
by FishBait on Thu Jul 21, 2005 at 06:56:23 AM EST

It seems that every Java programmer has come out of the woodwork to show off their works, so here is Yet Another Fractal Java Applet for your enjoyment.
This one is a bit different, it draws a fractal and allows you to zoom in, but can transform the data into audio data. I leave it for you to decide whether or not that is a good idea.

Fractals and uncomputables? (none / 1) (#95)
by expro on Thu Jul 21, 2005 at 08:18:46 AM EST

Perhaps this is unrelated, but I've been trying to get my mind around uncomputable numbers, especially since some seem to be definable and countable in some way.

There are aspects of fractals that also seem to defy computation, i.e. the closer you look, the fuzzier and more infinite the content of the thing you are computing looks.

Are these aspects of fractals ever related to uncomputable numbers?

Is it possible or useful to build fractals on a functions that produce uncomputable results?

No, not really. (none / 1) (#98)
by Coryoth on Thu Jul 21, 2005 at 01:58:40 PM EST

Are these aspects of fractals ever related to uncomputable numbers?


Calculating Pi, for instance, requires more and more work the further in (moe decimal places) you go.  Yet Pi is not a on-computable number, merey a transcendental number.


[ Parent ]

Must-read related book (none / 1) (#97)
by mbmccabe on Thu Jul 21, 2005 at 11:58:11 AM EST

Author's Link

Chaos - Making A New Science
James Gleick

With only a computer-geek's understanding of fractals (and a "Brief History of Time" understanding of space-time) I read this book with a good deal of enjoyment and fascination.

A very deep subject explained in easy language.

To quote the author's quotes only because I agree with them:

"An awe-inspiring book. Reading it gave me that sensation that someone had just found the light switch." --Douglas Adams

"This is a stunning work, a deeply exciting subject in the hands of a first-rate science writer. The implications of the research James Gleick sets forth are breathtaking."-Barry Lopez

I hope others enjoy it!


UltraFractal and fractal artwork (none / 1) (#111)
by the_idoru on Fri Jul 22, 2005 at 07:50:49 PM EST

It's unfortunate that there's no mention of UltraFractal. And, moreso, no mention of fractals as art. UltraFractal is basically the Photoshop of fractal rendering. Amazingly powerful. (Windows only, free demo available.) Another (windows only) fractal rendering software that is very popular among fractal artists is Apophysis, which renders "flame" fractals. Basically, millions of points mapped rather than the smooth color gradients that traditional coloring algorithms provide.

The artical is highly technical (and Part 2 sounds like it will be as well), and I find it a grave oversight to ignore the thousands of webpages out there where people have posted their fractal artwork. Really beautiful images, most of them made using UltraFractal. Hell, DeviantArt has a whole section devoted to fractal artwork. It receives close to 100 submissions each day. They go way beyond the lame psychadelic rainbow coloring algorithms that basic fractal rendering software provide.

Code it up! (none / 1) (#112)
by overcode on Sat Jul 23, 2005 at 03:46:36 AM EST

If you're a coder and find fractals kind of neat, try to write a Mandelbrot display program sometime. It's not terribly difficult, and it's wonderfully informative. The results are beautiful.

Making it run fast is the real trick. That's a worthy challenge.


Nope, (none / 1) (#117)
by Sesquipundalian on Tue Jul 26, 2005 at 06:41:02 PM EST

there's nothing skankier than the habits of a fractal coder who cut her teeth on Mandelbrot. Give me Brian's Brain coding nerd-hotties or give me death!

Did you know that gullible is not actually an english word?
[ Parent ]
The posthuman (3.00 / 2) (#113)
by Fen on Sat Jul 23, 2005 at 09:32:22 PM EST

On the other side, we may have a much better picture of what is truly chaotice. Being able to alter parameters by thought alone and experience the results through multiple senses will change how we view fractals.
Fractals Explained (none / 1) (#115)
by rs170a on Tue Jul 26, 2005 at 01:40:53 AM EST

Best explanation yet for bit-brained programmers like myself who struggled through math. Bravo.

Chaos Theory and Fractals in real life. (3.00 / 2) (#118)
by Saggi on Wed Jul 27, 2005 at 09:26:07 AM EST

Chaos Theory and Fractals in real life.

In the real world many systems behave accordingly to the Chaos Theories. These systems are often also fractal in nature, but this article is only aimed at the mathematical fractals.

A cloud is a fractal! It's in a 3 dimensional space, and the "rules" for it relates to the properties of water, air, temperature, pressure etc. that we find in our atmosphere. Just because the rules are not easily understood, and therefore can't be described in a simple mathematical equation, doesn't mean that Chaos Theory and fractal rules are cannot be applied.

Let me take an example that made me understand fractals:

The length of a coastline. For many years geography came up with different measurements of coastlines. Every time they tried to do it better, the coastline became longer... You may look in an atlas and see the different coastlines for various countries, islands etc, but they are all wrong? You need to understand fractals to describe a coastline. But lets me explain the problem.

If you move up to space, take a nice picture of an island, and start to measure the coastline you will find one length. Now you decent to a lower altitude. Here you use you high definition camera and gets a lot more details on you picture. Where you in the first image skipped tiny bays, they are now added to the length of the island. Finally you take a microscope, and start to measure the coastline length along the edge of each sand grain... this will give an extremely long coastline (except it will probably never be done in real life).

So how long is it? It's not possible to say exactly, because it's a fractal. But here the mathematicians have defined some helping tools. A fractal has a number describing just how "fractalised" it is. If it was a straight line it would not be fractalised at all, but these fractalisation may be calculated for various functions. Now if we could measure the coast line length and indicate how fractalised it is, we would now have a better way of describing this natural occurring fractal.

In real life we can't measure the exact fractalisation of a coastline, but we could do some approximations. A rocky coastline would have some average values, while a man made would be much smoother.


Real life fractals exist, and the use of fractal- and chaos theory might be useful in real life. Of cause only if you care... for most of us a coastline with fine white sand is better used for swimming or surfing, than studying math. I apologize if I have ruined your next trip to the water...


-:) Oh no, not again.
Fractals 101, Part 1 | 119 comments (81 topical, 38 editorial, 0 hidden)
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