Emergent Complexity

fun little mathematical curiosity called "cellular automata". This is one of the more blatant examples of a name being way too intimidating for the concept it embodies -- cellular automata are very simple, fun, easy to play with, math puzzles not much more complicated than a game of Battleship.¹ Think of a game of checkers, played on a big grid of squares (what they call a "checkerboard", to use the technical jargon), where the little pieces jump around and make each other disappear. Now, tweak the rules a little bit, start playing fast and loose with the rules. Say, give permission for checkers to reappear on the board, reanimated from the dead, in certain situations. Maybe, what the hell, make all the checkers just one color, instead of two opposing colors. Invent some different rules for when checkers have to be taken off the board, like maybe a checker has to go bye-bye if two other checkers wander up next to it. And that's basically it – a "cellular automaton" is just a big checkerboard grid, with checker pieces appearing and disappearing over time, based on whatever rules you want to make up for each turn.

That's it? Why the fancy name? Well, it turns out that if you pick your rules just right, all kinds of strange things happen on the board. Most of the time, nothing much special happens -- for typical sets of rules you pick, what happens over the course of the game doesn't look that interesting -- sometimes all the checkers 'kill" each other off until the board is empty, other times checkers multiply like rabbits until every single space gets filled and nothing else can happen. Occasionally, though, you stumble across a set of rules for playing the game where the checkers create intricate patterns that propagate across the checkerboard like ripples on a pond.  The patterns move around on the board, interact with each other, spawn new patterns -- in effect, start to look temptingly like little life forms.

By far the most famous of these is a rule set created by John Conway, called the "Game of Life" since it makes such startlingly lifelike creatures. Here's how you play the game:

• Start with a normal game of checkers, but play with only one color. There's no competing white vs. black.
• Throw down some checker pieces at random, wherever you want.
• At each turn, look at the pieces on the board, and decide which piece lives or dies according to how many neighbors it has. If a checker has no neighbors or just one neighbor, it "dies of lonelieness" and is taken off the board. Four or more neighbors, and it develops an intense hatred for life in the crowded slums, and it leaves the board. Two or three neighbors, and the checker is happy, and stays on the board.
• For any empty space on the board that has three neighboring spaces with checkers, you get to reanimate a dead checker, and put it in the empty space to wander the earth as an undead zombie checker, forever doomed to feed on the brains of chess pieces.

It looks kind of boring, might get to be a pain in the ass after a while to be counting neighbors. But if you code it up on a computer, and let the computer work out all the details, you can just watch the game go by in real time. If you do this, you'll see all these funny little shapes made from clusters of adjacent checkers, moving around the board and bumping into each other. A bunch of shapes are possible -- they've been named "gliders", "spaceships", "boats", based on their shape and what they seem to do. There are even "guns", persistent shapes that burp out gliders and other shapes.   Check out the example shown here (borrowed from the excellent Wikipedia page).  It's frankly startling the menagerie of creatures that inhabit the Game of Life.  Lots of really smart people spend their free time exploring the world, cooking up interesting examples, and generally competing at out-nerding each other with what they can get the game to do.   

Chaos Theory (a division of physics that has the all-time coolest name of any theory), where random-looking complicated behavior pops out of extremely simple situations. There's no question that this area is very interesting, poorly understood, and potentially very important to understanding the world, but I think Wolfram's let it get a bit to his head.

Sooner or later, anyone who studies physics gets seduced by chaos theory. I won't go into detail on it here, but the cool aspect about chaotic systems is how they can create complex, intricate, random-looking patterns from very simple scenarios. For example, a simple little mathematical equation called the logistic map is so easy to work out that you could compute it on a watch calculator (remember those?)², and yet it spits out a string of numbers that look completely random. They *look* random, but they're not -- if you spend the afternoon cooking up the first 1000 numbers from the logistic map, and your eighth cousin Helmut in Germany does the same thing next week, you'll both get exactly the same numbers. Compare this to the last 1000 lottery numbers that popped up in the North Dakota state lottery -- these numbers really *are* random, and couldn't be predicted or duplicated by any process, much less something as simple as the logistic map.

And here's the tempting bit. You occasionally stumble across simple little procedures (such as cellular automata or the logistic map) that generate very complicated, random-looking patterns. You also happen to see complicated, random-looking patterns all around you every day in nature -- in the swirls of milk in your coffee, the pattern of stars in the sky, even the last 1000 lottery numbers in the North Daktoa lottery. So... any chance that some of these are being created by simple little setups like a cellular automaton? Could something that looks random (like the stock market, for example) really be completely determined by some unknown but simple little equation, that we just don't happen to know? Maybe the randomness of the world would resolve itself, become predictable and understandable, if only we could discover what simple mechanisms are creating it all.

It's a tempting bit o' reasoning, but so far there's no real evidence that it's true on a grand scale -- things out in the world that look complicated and random might really be just complicated and random.