The Sheila Variations

Well, uhm, you're asking a big question.

The big thing to understand about fractals is that you have these equations that, instead of resolving, constantly result in a new answer that, when you repeat it, just results in more answers, and never any resolution. So you get answer after answer after answer, that sort of builds on itself, although somehow you wind up with these recognizable patterns anyway.

Gee, that was non-specific an answer, wasn't it?

So, uh, I mean, did you have any specific questions? :-)

I always hated fractals. All that hard math converting 1/2 to 2/4 so that you could add it with 3/4...

Oh, wait...

And - do fractals actually manifest in the universe, on the physical plane? Or - is it just all in the math?

Oh -

and who "discovered" fractals? It's a relatively new concept, right?

So - what exactly is the purpose of them?

Or is it just a weird series of formulas some mathematician discovered?

I believe the Greeks had the first implementation, based on geometry, and used heavily in building and architecture.

Also a handy source of making fun of europeans. Like so:

The metric system is for people who can't divide.

:)

They're freakin' awesome. I suppose it's possible that someone would need to know more than that, but that's all I've ever needed.

Nathan -

"freakin' awesome" is about as far as I have gone as well!

But now I need more. More!!

As I understand it, a fractal pattern is one in which, the more you zoom in on it, the more detail you find. The pattern never resolves itself into simple things, no matter how small a part of it you look at..there is always more complexity.

An example often used is that of a coastline. If you look at it on a map (say, of the east coast) you will find sinuosities--bays, inlets, etc. But if you look at it on a more detailed map--say, a topographic map covering 10 miles--sinuousities will appear that weren't there on the larger map. Thus, if you measure the "length" of a coastline on the east coast map by wrapping a ruler around it. If you do the same with the smaller-area maps and add them together, you will get a *different answer* (always a larger number). And, in fractal principle, this continues all the way down.

Do fractals exist in reality? It seems to me that approximations to a fractal exist (like the coastline) exist, but a pure fractal cannot exist, because it would embody an infinite amount of information. But I'm not any kind of expert in this stuff...

David -

Okay. Another reader just emailed me, explaining the coastline metaphor too.

I like that a lot. I can understand that.

Would a flower of frost on a window be another example?

i recommend The Science of Fractal Images, by peitgen and saupe, eidtors. there is just about everything you need to know in there.

The man who I would think of as the "father of fractal geometry" is Benoit Mandlebrot. Basically, a fractal can be thought of as geometry that results by iterating a process infinitely often. As for 'occuring in the real world' they can't quite do so because the real world is quantized. you can't keep breaking an atom down forever. But they provide a better 'model' of reality than non-fractal geometry does. To get an idea of the process involved, start with a straight line

___________________________

Then in the middle third part, replace it with two lines at an angle like this

/\
/ \
/ \
===== =====

Then repeat that process for each line segment, and then keep repeating ad infinitum.

crap, sorry about that, the blog turned my nice picture into something meaningless by undoing the spaces. Go to this website for a nice interactive example.

To answer basically, YES, it ABSOLUTELY maps to reality. An ongoing debate in the scientific community--well, it's not really so much a "debate" as a point of discussion--is where fractals are descriptive, and where they are not. Where the apply, and where they do not.

Once you see enough fractal graphics, you start to see amazing patterns that repeat themselves over and over within nature. One series of equations in particular that I've seen are amazing. You run them, and watch them go, and suddenly you see a shape that looks like a human womb, a coastline, and and a cluster of stars, all at once. Or another series, and it looks like a supernova, a human hand, and a splash of ink.

This is what makes them so cool. Where does reality match these non-linear equations, and where does it not? Apparently, they match a lot of things.

By the way, fractals are a branch of "chaos theory," which you must have heard of. That whole "a butterfly flaps its wings and causes a hurricane 500 miles away." This is the SAME EXACT concept as fractals. You have this tiny little equation, but one answer builds on another and builds on another, and it's IMPOSSIBLE to predict where it's going exactly until you see the results, and then in retrospect you look and the pattern seems rather obvious, if breathtaking.

Which points to the pitfall of the whole thing: not everything works that way. Not every butterfly causes a typhoon. If small effects caused such huge consequences all the time, then we probably would have knocked the Earth out of orbit by now just from jumping around on it. SOME things in the universe do NOT work by the rules of these non-linear equations. SOME do. Where do they apply, and where do they not? WE DON'T KNOW.

We are now pretty sure, however, that whole big things about the universe appear to work like non-linear equations. Others seem to work along the lines of classical Newtonian/Einsteinian style physics, with everything linear and neatly solved, like the stable orbit of a planet around a star. Others, like the genetic sequence that makes you you, well, even identical twins are not completely identical are they? One's gay, one's straight, what happened? Something somewhere did. Probably something very small.

It's a fascinating subject that raises as many questions as it provides answers.

In fact, since I'm on a roll:

Fractals and chaos theory in general seem to point to a resolution to a problem that troubled Einstein.

One of his most famous quotes was "God does not play dice with the universe." Now that was in response to Quantum theory, which repelled him, because it had (and still has) a fundamental uncertainty built into it that can never be resolved.

Einstein, Newton, and other classical physicists believed that EVERYTHING in the universe must ultimately fall into a predictable series of equations. "If I strike the pool ball at this angle with this much force with this size stick, then I will know which balls fall into their pockets and which do not, and I will know it before I even strike."

It even goes to the concept of predestination, which is a bugaboo for theologians and philosophers. If EVERYTHING works along mechanistic lines, then there CAN be no free will. You, Sheila, are nothing but a collection of atoms making up a massive collection of molecules, but if we knew EVERYTHING about the universe's conditions from the beginning, and had a big enough calculator, we SHOULD be able to predict your eventual conception, birth, and death, and everything you ever do, think, or say in your entire lifetime. You're nothing but a difficult problem that God had solved before he threw his hand out and created the universe. If we knew enough data in advance, and our math abilities sufficient, then we could easily predict that at this point in space and time you would have posted this question on your weblog, and I and all the rest of us would have written this answer for you.

Quantum mechanics, on the other hand, posits a certain random nature to the universe that makes prediction a dicey proposition. At any given split-nanosecond, there are two possible ways things could go, and the result is more or less random. Broad predictions can be made, but the more precise you try to get, the more likely you are to fail.

Chaos theory (which, again, fractals are part of) is a fascinating bridge between these concepts. Becuase you cannot predict the end results of a non-linear equation. You can only solve, and solve, and solve, and repeat ad infinitum and see the result. AND AND AND, here's the weird part:

You can't resolve them backwards and get your original starting point.

In other words, you can start at a certain state in the series of answers, and try to work backwards--but when you do, you almost never wind up at your original starting point. So you can't just say, 5+5+2+3=15-5-5-2-3. With these non-linear equations, you cannot look at your result from a certain state, and work backwards to get back where you started. Instead you wind up somewhere completely different.

See what I mean?

Final thought: for thousands of years, mathematicians and physicists ignored non-linear equations, mostly because they found them irritating and assumed that the universe couldn't possibly work that way. Also because you could spend a lifetime answering them and never resolve to anything.

It was only when someone (Mandelbrot) thought to use a computer to start experimenting with them, and graphing them out that way, that suddenly he realized, "Whoah, look at that."

Quantum physics and chaos theory are very dear to my heart - all of this is music to my ears.

To me, it makes sense on such an ultimate gut-level, but i could not tell you why. Perhaps because my own personal experience of life is relatively non-linear ... and so the scientists seem to be touching on some kind of truth. Some kind of human truth, I mean.

Ron -

The name Mandlebrot is familiar to me. I believe it was mentioned during a marathon date I had a couple summers ago, where the guy and I drove to Atlantic City on a whim, in the middle of a rainstorm. I didn't like the guy that much, but he did know a lot about Mandlebrot. And it was a long long drive.

I will do a bit of research on Mandlebrot ...

This is all great stuff.

It's not just non-linear equations are 'irritating'. Non-linear equations (by which in this area people mean non-linear differential equations) have no general technique of solving.
Occasionally certain types have "tricks" but that is the exception rather than the rule. In reality, the only good way to solve them is to approximate numerically. Without computers this was nigh impossible to do. The relationship between these and the popular notion of 'chaos' is that non-linear equations are 'sensitive' to initial conditions much more strongly than linear ones. In other words, a small change in the initial condition can result in a vastly different solution. Thus, you get 'chaos' because initial conditions can't be replicated. This is the idea of the so-called 'butterfly effect'.

I'm a little late in all of this, and Dean has given some nice summaries aready. I would only add that Chaos: Making a New Science
by James Gleick was an excellent book that tells about chaos theory, fractals, and the history of discovering both, while generally keeping things easy to understand on not getting terribly math intensive.

Aaron -

Never too late. I've written down what everyone here has said, and I will add your book to my list. Thank you very much!

Just to add to what others have said, fractals do relate to the real world.

While such things as the layout of seeds in a sunflower head are related to the Fibonacci sequence, things like branching patterns in trees and other plants are often fractal.

Clouds can be fractal (in the book Chaos, Gleick profiles a researcher who lost his air travel privileges because he was spending so much time in the air looking at clouds).

Line noise on telephones has been shown to follow a fractal pattern (the Cantor fractal, IIRC).

Fractals are proving useful in earth science (geophysics), materials science (crack propagation), and probably many fields I haven't heard of.

And the graphics programs are fun!

The James Glieck book is terriffic. When it came out I was a security analyst and our head of research bought a copy for everyone in the department. I have found it useful at least in a qualitative sense in thinking about several new medical technologies including:barbed suture material,bone grafts and sustained release drug formulations. I understand it is also useful in understanding metal stress and other aspects of material science. You wont be able to pur the book down--

Gleick's book is one of the best for a non-mathematician. Be careful with websites that oversimplify. One of the most common errors is to equate Chaos and Heisenberg. Quantum mechanics contains an Uncertainty Principle based on very different mathematical principles from those that underlie non-linear dynamics. If you want a decent and not too technical explanation of the math behind the Uncertainty Principle, see www.numberwatch.co.uk/uncertainty_principle.htm

Another error made by popularizers of non-linear dynamics is to attribute chaotic systems solely to non-linear equations. Just because an equation is non-linear does not mean that the relationship between input and output is seemingly random (as is the case for truly chaotic systems). For example the equation y = x^2 is non-linear, but not chaotic. Many mathematical chaotic systems are built on differential equations, but not all non-linear partial differential equations are chaotic either.

One of the features of many real chaotic systems is feedback - the butterfly flaps his wings and causes a storm, but the wind created by the storm far away eventually rebounds back to the butterfly's home and blows her off of her original course, which changes the location of the next storm she causes.

Okay, minor nitpick time. The fact that not all non-linear systems don't lead to chaotic responses doesn't mean you can't attribute all chaotic behavior to non-linear equations. You're right of course that you can have very non-chaotic nonlinear differential equations. But a linear differential equation will not have chaotic behavior. While I don't claim to be an expert, the key seems to be sensitivity to initial conditions. If small changes in initial conditions to a differential equation lead to very different solutions, that is considered chaotic. Systems with feedback tend to have this since every pass through amplifies the change.

Wow - all this learned talk about fractals makes my head hurt. They didn't seem that complicated to me. A fractal is a structure with an infinitely repeating pattern. No matter how much you magnify (or expand) your view of them, the pattern is the same. You can express this mathmatically, which can be really complicated or really simple (depending on how complicated the pattern is), but it seems a lot easier to understand if you just draw them. I guess that was Mandelbrot's genius.

Ron - I think we are in violent agreement. I just have no patience for people who talk down to laymen about "chaos" by explaining the concept as if non-linearity were the only pre-requisite for chaotic behavior. It's sloppy thinking, since as you point out, chaotic systems are a subset of non-linear systems and other conditions also apply. The same goes for systems with feedback.

I used to teach chemistry at the college level, and a lot of TAs would teach the Bohr atomic model as if it were gospel because it is easy to grasp, which would confuse the heck out of the students when the concept of orbitals was introduced. I get a little upset by that kind of condescending behavior, which I found all over the web in discussions of fractals.