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Isn't that cool? The key thing about fractals is that if you look at just a small part of it, it resembles the whole thing. For instance, the following picture was obtained by zooming in on the upper left tail of the previous one:
One of the original "big ideas" of complex systems is that fractal patterns seem to appear spontaneously in nature and in human society. Let's look at some examples:
Physical Systems: Pop quiz: is this picture a close-up of a rock you could hold in your hand, or wide shot of a giant cliff face?
I don't know what the answer is. Without some point of reference it's very hard to determine the scale because rocks are fractal: small parts of them look like the whole.
Other examples in physical systems include turbulence (small patches of bumpy air look like large patches) and coastlines (think Norway). These two examples in particular inspired Benoit Mandelbrot to give fractals their name and begin their mathematical exploration.
Biological Systems: Here's an example you're probably familiar with:
And one you probably aren't:
The first was a fern, the second was a vegetable called a chou Romanesco, which has to be the coolest vegetable I've ever seen.
In the case of these living systems, there's a simple reason why you see fractals: they are grown from cells following simple rules. The fern, for example, first grows a single stalk with leaves branching out. These leaves follow the same rule and grow their own leaves, and so on.
Of course, the pattern doesn't exist forever. If you zoom in far enough, eventually you see leaves with no branches. This is an important feature of all real-world fractals: there is some minimum scale (e.g. the atomic scale or the cellular scale) at which the fractal pattern breaks down.
Social Systems: Some people like to extend this reasoning to the social realm, arguing that individuals form families, which form communities and corporations, which form cities, nations and so on. You can try to draw parallels between behavior at the nation level or the corporation level to behavior at the human level.
Personally, I'm a little dubious on this argument. My doubts stem partly from my personal observation that humans seem to act morally on an individual scale, but that corporations on the whole behave far worse than individuals. I think there's something fundamentally different about the centralized decision-making process of a human, and the more decentralized process of a corporation. But this is all my personal opinion. Feel free to debate me on it.
From one perspective, the idea of "emergence" is that systems are not completely scale-invariant. Add up a bunch of pieces, and the aggregate displays properties and behaviors which the individual pieces don't. My favorite weird example lately has been Ian Couzin's flocking simulations: each bird has a simple rule for what to do, and individual birds have no memory, but the flock as a whole exhibits a hysteresis effect, remembering where it's been.
ReplyDeleteThe more surprised we are by the properties of the whole, the "stronger" we say the emergence is. Sometimes, the aggregate behavior can be predictable from the basic laws of the small-scale parts (e.g., temperature and pressure of a gas), but in other cases, there's so much historical and biological contingency involved that the basic laws of quantum mechanics, electromagnetism and such aren't sufficient anymore. The ability of chlorophyll to trap sunlight for photosynthesis is an example: we can check that the reactions are consistent with physical law, and we can use physics to calculate various things about the phenomenon, but we couldn't start with quantum mechanics and an empty notebook and then prove a theorem saying that photosynthesis will happen.
On the topic of corporations and how these "artificial legal persons" differ from real persons, John Baez has some stuff.
"Pop quiz: is this picture a close-up of a rock you could hold in your hand, or wide shot of a giant cliff face?"
ReplyDeleteI experience a practical consequence of this all the time when I skydive. Because of the scale-invariant nature of clouds, it can be very hard to determine the altitude of a cloud layer by eye. Is it a big cloud formation far away, or a small formation near the ground? Who knows?
@blake-Maybe the right thing to say is that fractals are a particularly simple kind of nonlinearity. In linear systems like ideal gases, local interactions average out to boring behavior on a large scale. The systems we call "complex" are nonlinear, and fractals are an example of nonlinearity that arises when there is some recursive process going on. So maybe fractals are so ubiquitous precisely because they occur in the simplest nonlinear systems.
ReplyDeletePhotosynthesis obviously requires a much higher degree of complexity than pure fractal systems possess.
@gg-Wow, that sounds really scary: to be falling through the air and not have any idea of the size of objects rushing past you. Good thing the ground isn't completely fractal, or else you'd have serious trouble knowing when to release your parachute.
Good post, love the fractal vegetable!
ReplyDeleteRegarding social fractals, I think that the concept is only a mildly useful one. More relevant is are the concepts of cultural agency and coherence.
Fractals are pure mathematical structures, and seem better at describing physical systems than biological, and better for biological than social. I.e. the farther up the levels of organization you go, the less good the fit between the fractal model and the system in question. Which is to say, I agree with your claim :-)
I remember this phenomenon from my previous life doing 3D modeling and textures for video games.
ReplyDeleteI had a working theory that I could save on texture space by reusing the exact same texture for multiple levels of detail, based on the fractal nature of organic things.
As in, I could use on texture to show a plant from a distance, then as the camera moved closer, shrink the same texture and use it again for the detail.
It worked very well for trees, dirt, and rocks. I've always loved the way trees and plants seem to grow from a simple algorithm.