Wisconsin Public Television University Place  Self-Organization: Nature's Intelligent Design  Clint Sprott  7/10/13 UP897  cc >> Good evening, everyone. I'm Cassie Immel from the Biotech Center. This evening, we have Clint Sprott from UW Madison. He's an emeritus professor in the physics department. He earned his bachelor's degree from MIT in 1964 and his PhD in physics from here at the UW. His professional interests are experimental plasma physics and nonlinear dynamics. Currently, he is the mind behind the Wonders of Physics shows. So, every year, at the Physics Fair, Wonders of Physics has some great shows going on on the weekends. Great for families and children of all ages. And the Wonders of Physics has actually been at other events across the UW Madison campus and all over the world. 

So, without further ado, let's welcome Clint Sprott. [APPLAUSE]  >> Thank you, Cassie. Now, despite the title of my talk, this is not going to be a talk about religion. [LAUGHTER] Although, I might make a comment about that at the very end if there's a little bit of time left. Rather, what I want to talk about is the way in which complicated systems in the world arise out of what seemingly is a random starting condition. And I'm going to show you, actually, six examples of this, and they're all examples that were programmed on the computer and they were all cases that I've worked on to one degree or another. The first example is right here on this slide. This object, which looks like maybe something you might see on your windows when ice forms in the winter or it may look some species of lichen or something, is actually something that is called diffusion limited aggregation. And I can describe to you how something like this might come about. Imagine this room was filled with tiny dust particles floating around, and one of those dust particles was sticky. 

And, after a little while, another dust particle collides with it and sticks to it, and it then becomes sticky. And this process goes on over and over and over, and after a long time, a pattern that looks like this grows. Now, this one was obviously simulated on the computer, but it illustrates an example of how in nature a complex pattern, like this, may arise from simple random rules. There's no one dictating how that should form. It's formed by the random collision and sticking of these individual dust particles. Now, lest I get too far into this, let me point out to you that there are a lot of examples in nature of systems that have apparently self-organized. I show you, on the upper left here, the most distant photograph ever taken. This was taken by the Hubble Space Telescope looking at a region of the sky where there are no nearby stars. And so everything you see in this picture is a galaxy. They're far away and there are many of them. 

It's been estimated there are between 10 and 100 billion galaxies in the universe, each one of which has between 10 and 100 billion stars. >> How do they know none of those are stars? >> Well, they know they're far away because the light from them is strongly red-shifted, and that's an indication that they're very distant. Also, they're looking in a region where there are known not to be any nearby stars. So, this is an example of how, on the very largest scale, nature has self-organized. The matter in the galaxy has clustered into these groups that we call galaxies. And we, of course, live in one of those: the Milky Way galaxy. Now, if we get closer to the Earth, on the lower left here is a picture of the Earth seen from a geostationary satellite high above the Earth. And you can see weather patterns evidenced by the cloud formations here. And you can see that the clouds seem to have some degree of organization to them. They're not scattered around randomly, but they seem to be trying to do something. 

If you get a little closer to the Earth, on the lower right here is a picture of the United States taken from Earth orbiting satellites, presumably on a succession of nights so that you can get rid of the clouds. And what you see are the lights from the ground, and the lights are sort of a surrogate for where the people are. And when you look at this picture in the lower right and compare it with the one in the upper left, you see a striking similarity. And you may ask yourself: why is it that people collect into groups in a way very similar to the way matter in the universe collects in the galaxies? Obviously the thing attracting people together and the thing attracting matter together are two very different processes. One governed by Newton's laws of gravity, and the other governed by some kind of social dynamics. And yet the end result is strikingly similar in the two cases. If we get a little closer to the ground in this upper center, this is a picture of the landscape over some portion of the Earth, I don't even know where it is, but you see mountains, rivers, coastlines, lots of intricate features that somehow organized themselves. Now, the one in the upper right you may not recognize, but that's actually a slice of the human brain. Not my brain but someone's brain. And these dark objects are neurons, and they're connected by synapses and axons which you can barely see in the picture. 

And that connection, by the way, is similar to the connection you see in the lower right. If you look very carefully, and probably the lights are a little too bright for it, but you can actually see the interstate highways that connect the population centers. And so there is a connection and it is also known that in the galaxy, the galaxies are distributed in a way such that there are long connections between particular galaxies. They're not distributed uniformly throughout the universe. So these are examples from the largest scale of the whole universe to the microscopic scale of the brain of examples of systems that are self-organized that have clustered in some fashion to achieve some goal, presumably. Now, I showed you the most distant picture ever taken, this is the oldest picture ever taken. This is what the universe looked like about 400,000 years after the big bang. It sounds like a long time but if the universe was a 70-year-old person, which is about my age, this would be what I would look like one day after I was born. So it is a time very early in the life of the universe, and you can see a lot of interesting structure there. But that's exactly the wrong message to take away from this. 

In fact, this is a photograph, not in visible light, but it's a photograph in microwaves because after the big bang lots of light and photons were produced, but they've been cooling down in the 14 some odd billion years that have occurred since that. And now they've cooled down all the way into the microwave range. And that's what we see today. It's the remnant of the big bang. And it looks like there's a lot of structure, but in fact what's shown here are the variations in temperature over the entire sky, and the variation from the deepest blue to the brightest red amounts to about one part in 10,000. So at this very early time, the universe was extremely homogeneous. Very, very little structure to it, but it was very hot and that's important. And yet, out of these small variations, a hundredth of a percent in temperature, grew the galaxies. And so these tiny, tiny variations in fact were the seed for the self-organization that proceeded to occur. Now, I became interested in self-organization about 10 years ago, and I encountered some forest management people here on the campus, right across the street in Russell Labs, and they were analyzing the data that had been collected in the mid-1800s. 

At that time the government wanted to sell off a lot of the public land, and they sent surveyors out to survey the whole United States east of the Ohio, or sorry, west of the Ohio River. And the surveyors went and on a one-mile grid they recorded what they found, what kind of landscape, whether it was a forest or a prairie or a swamp or a lake or whatever. And about a dozen years ago or so, these data that were carefully written down in journals were digitized, put on the computer, so that one can examine them in greater detail. And when I entered the picture, the forest people were curious about why the red areas, which represent forest, were where they are and the green areas, which represent savannas, are where they were. And they were asking questions. Maybe it has to do with the soil or the moisture or the climate. But I had the gut feeling that maybe they were asking the wrong kind of question. Maybe there is no reason for this any more than there is the reason that the Earth is 93 million miles from the sun and not some other distance. Perhaps it just kind of happened that way. So I began to ask what is the simplest kind of model that would have the possibility of producing that kind of pattern. 

And the model that I was looking at at the time for a totally different purpose is something called a cellular automaton. And this particular one I'm going to describe is stochastic. That's a fancy word that simply means random. It means that this model is going to have some coin flipping involved, some random choices. Well, the simplest way I could think to produce a pattern like that is to first start with a grid, sort of like a checker board. And so here on the upper right you can see these points on this grid, and those would represent where the surveyors went and sampled the landscape. The simplest model I could think of was that at random times whatever you found on one of those grid points you would remove it and you would replace it by something randomly chosen from its neighborhood. And so the neighborhood is taken as a circle, and the particular example shown here, the circle is a little bigger than, if these were on a one-mile grid, a little bigger than a mile. And that circle takes in four other points, one north, one south, one east, and one west. So you roll the dice, and you decide which of those four neighbors your side is going to be in the next generation. 

Well, you do this not just once but you do it over and over and over and over again, millions and millions of times, replacing all the sites at random times with randomly chosen neighbors. Simplest thing I could think to do. Now, there's some technicalities involved in this. For example, what do you do with the boundaries where the radius extends outside the grid? How do you start it? You have to give it some initial conditions. How do you keep it from evolving all into one form of landscape or another? These are technicalities that are not really relevant here but were important in actually implementing this. Now, this is a so-called one-perimeter model in the sense that the only thing you have at your disposal to adjust to make it fit the data is the radius of the neighborhood. You take a bigger radius, then more of your neighbors are candidates for what you become in the next generation. Well, you can run this model starting with different kinds of initial conditions. And I show you two extreme examples of initial conditions. The one on the left is a totally random initial condition where every site is given one of those six landscape types, indicated by the corresponding color, but they're distributed randomly but in the correct proportion. 

The one on the right is an extremely ordered landscape where all of the trees are put down in the south here. By the way, this is northern Wisconsin. You can see Door County up here, and this is Lake Superior up here and Lake Michigan over here. And so this is a highly ordered initial condition. The interesting thing is, after you run this model for a long time, it doesn't matter which initial condition you chose, they will both settle down to the same kind of pattern. A sort of surprising result. In one case, you could say you started with a highly disordered situation and it became more ordered, if indeed that's what's going to happen. The other case, you started with a highly ordered initial condition and it becomes more disordered. And so, the real world lies somewhere at the boundary between order and disorder. Well, I can actually show you, I'm actually going to run the code here. 

Whether this will come out on the television, I don't know. But the computer program is actually running right now, simulating this, starting with the random initial conditions. So on the right is the data that we're trying to replicate. On the left is the initial random conditions that are put in. And you can as time goes on, and here's time down here going forward in logarithmic steps so it actually goes through about 8,000 generations, 8,000 replacements, you can see it develops a landscape that is not unlike the one that's observed artificially. Now, of course, you don't expect the reds to be in the right place and the greens to be in the right place. There's nothing that would have done that. In fact, that was the reason that I did this in the first place was to try to convince people that there's nothing special about where the reds and the greens are. They could be anywhere. But there is a perimeter to the model, and in this case it was chosen to have a value of about, somewhere it's down here, R is three. 

And so it's a three-mile radius. You look at all the sites within that radius, a couple of dozen sites, and you choose one at random. If you choose too small of a neighborhood, you get too much self-organization. If you choose too large of a neighborhood, you get too little self-organization. So there is a value of the perimeter that gives the right sort of picture. And this is quantified by, there are many ways to do this, by something that we call the cluster probability which is simply the probability that a given site is the same as its four nearest neighbors. Or to say it a little better, that your four nearest neighbors are the same as you. That will mean that you are part of a cluster. And so, in the real landscape, there's about 30%. You can see this evolving toward that 30%. 

In the lower left, you can see a graph showing it approaching that, and you can approach it either from the random side where there's very little clustering or from the ordered side where there's a lot of clustering and the same kind of behavior will occur. Now, I went around the country talking about this model, and I was kind of proud of it because it was the first thing I thought to do and yet it gave such beautiful agreement in a sense. And when I was up at the University of Minnesota, I gave a talk on this, and there was a young woman mathematician in the audience and she came forward and she said, oh, that's the voter model. I said, voter, I never heard of voter, who's voter? She said, no, no, no, it's a model that people have used to predict elections. Do you recognize this picture? >> Red and blue states. >> Red and blue counties actually in the 2012 presidential election. McGovern and, or not... [LAUGHTER] Who ran? The Obama and Romney... [LAUGHTER] Election map. >> You're 40 years behind. 

>> I'm showing my age. So, indeed, this is a model that had been studied by other people. So I reinvented it, if you will, so I don't get a lot credit for that. But anyway, it was an interesting example of two processes in very different fields, trees and political parties, that somehow organized themselves in a very similar way. As if there is some deep underlying connection that's not at all obvious between the two. Now actually, when I did this, it was not 2012, it was back in 2004, and that was Bush and Kerry as well as I remember. The maps look almost identical, right? Perhaps not surprisingly because not too much time has lapsed between the two. But I was interested in whether one could generate a pattern like that with this voter model as has been claimed to be the case. So here's the voter model. This time on the upper left I'm starting with a highly ordered initial condition where I give Kerry everything north of Minneapolis. 

[LAUGHTER] Bush everything to the south, and then I let it evolve in time over many generations and look at the cluster probability and adjust the single perimeter R, in this case it's about 1.6, to get a similar cluster probability of about 70%. And here you can see an example of a highly ordered system becoming more disordered as time goes on and approaching this universal state where it seems to be stable. And of course there's no reason to expect the reds and the blues to be in the right place. There's absolutely nothing in the model to accomplish that. But it does achieve the correct clustering. So in some sense it is predicting the way in which people organize their political views. In the simplistic model, one would simply say in this election you're going to vote the way one of your randomly chosen neighbors voted in the last election. And if you have everybody following that simple random rule, you discover that you develop pockets of democrats and pockets of republicans and you eventually are persuaded, I suppose, by your neighbors, not necessarily your geographical neighbors but the people that most influence you, to vote in a particular way. Over the course of time, you do that. Now, you notice this pattern is not static. 

When it evolves to the right cluster probability, it is continually changing even as are the trees. So it's not a static state that it goes to but it's a dynamic state but it's one in which certain characteristics, such as this cluster probability, seem to be preserved as time goes on, at least in some average sense. Question? >> Yeah, how can you be measuring? I guess my question is, you're going back 8,192 generations, there weren't that many voters then, so how do you come up with these measurements? >> Well, that's a good question. 8,000 generations, does that mean 8,000 elections? We weren't even around 8,000 elections ago, but of course we didn't all start from the highly ordered initial condition either. We presumably started with something close to the final state. So it probably took a lot less than 8,000 generations to evolve into this condition. Maybe very little evolution was necessary. Maybe it started out very close to its final state. But, of course, this process can go on generation after generation of influencing one's friends. You may notice that this distance of 1.6 takes in about 10 or 12 or so of your neighbors. 

And if you think about how many people really influence your political persuasions, probably something of that order. Probably roughly that many people whose opinion you respect enough to have it influence your vote. It's the right order of magnitude. It's a number bigger than one but less than hundreds. Well, so, okay, I had a great idea, but I wasn't the first to have it. Now, both of the models that I've shown you, the diffusion limited aggregation, the dust particles moving around, and this voter model are stochastic models, which means that there's randomness built into it. The dust particles are moving randomly, and one's flipping a coin to decide which of the neighbors to choose. I got interested in whether there were deterministic models that would do the same thing. What if there was no coin flipping involved and there was a specific rule that tells you you look at a specific number of your neighbors, let's just say your four nearest neighbors, and then there's one and only one thing you can do depending upon their state. That would be a purely deterministic model, and your gut instinct and the gut instinct of most scientists would be that in such a situation nothing is complicated as what I've been showing you could occur. 

Such is not the case. Let me tell you about a deterministic cellular automaton. First of all, why would one want to do this? Well, there are a number of reasons that deterministic models are preferred over random or stochastic models. The first is that randomness is a way of concealing our ignorance. When we don't know what else to do, you just randomly choose, right? If you had a real good model for what's happening, politically or to the trees or in other circumstance, you would use the dynamics that's governing things and you wouldn't be choosing randomly. So we would like to do better than to just sweep our ignorance under the rug. Secondly, one of my goals in this was to convince the ecologist that simple models could produce complex behavior. Most people's gut instinct is that when you have a complex behavior, like the voters or like the trees, there must be a complex underlying cause. Well, if you have randomness in a model, there could be all kinds of complexity buried in the randomness, but if you have a simple set of rules, then it's a lot harder to argue your way out of that being obvious that it should be like that. So simplicity can produce complexity. 

Simple rules can produce complex behavior. The third thing comes back to a personal interest I've had for the last 20 years or so, and that has to do with something called chaos about which I'll say a little bit more later, but chaos is the apparently random behavior of deterministic systems. So it can only occur in a deterministic system, and I was interested in whether these systems had that property, and indeed they do, not surprisingly. And finally, if you have a set of rules, rather than just flipping coins, the rules might provide you some insight in what's happening in the real physical world, in the real situation you're trying to understand. So these are reasons one might want to do that, but of course, when I did it, I wasn't conscious of having these reasons and I generated them only later when somebody asked me, hey, why are you doing that? Well, how does a deterministic cellular automaton work? Let's just consider a cell in the middle, which I've labeled zero here, and let's just think about its four nearest neighbors, north, south, east, and west. Well, let's even further simplify and say that each cell can take on one of two states. It can be either dead or alive. It can be either a tree or a prairie. It can be either a democrat or a republican. Gross simplification of the real world, but interesting things happen even with that degree of simplification. 

Now, one needs a set of rules. You're not going to be able to evolve this randomly, but you have to have rules. So let's focus on the cell in the middle, and let's say that's you, and you're going to look at your four nearest neighbors and decide what you're going to do next, whether it's an election or whether you're a tree that's going to die or whatever. Well, there are many rules one can devise, even with as few as four neighbors, but one rule is to look at your four neighbors and ask whether they are all in the same state, and if not, how many of them are in a particular state. So you could count up the number of them. Let's just say they're all four republicans. So there would be, in this so-called totalistic rule, you could be either a democrat or a republican, so you could be either a zero or a one, but you could have either no neighbors that are democrats or you could have one, two, three, or four neighbors that are democrats. And then, for each of these possibilities, you can either change you state, change you mind, or you can stay as you are. You can be a democrat or you can be a republican as a result of interacting with these four neighbors. And it looks like there's 10 different possibilities. 

But, in fact, there's more than that. There's two to the 10th possibilities because each one of these states involves a binary number. So this is a 10-bit binary number. So there are, in fact, two to the 10th possible rules you could come up with, even for this very simple model where you have four neighbors, and you're just counting the number of them that are in a particular state to decide what to do next. Well, that's over a thousand different rules. Now, most of these rules don't do anything very interesting. If you set up a cellular automaton and run it, the whole system will just evolve into everyone being a democrat or everyone being a republican or you may get periodic oscillations or things that are relatively simple but you tend not to get those interesting landscapes. Now, of course, this is a very simple rule. Suppose you look at your 20 nearest neighbors. Well, that's not a lot of neighbors. 

You probably have that many people that influence you from one time or another, your politics. Well, a number of rules is two to the 2,250. It's an astronomical number of rules. And you would think that most of these rules don't lead to anything interesting and finding the ones that lead to interesting behavior is like finding a needle in a haystack. Well, it is a little like that except there's not one needle in the haystack, there are millions of needles in the haystack, even more, and so the computer is good enough that it can find the interesting rules, even out of this extremely large set of possibilities. And I'll just show you one simple example. The one on the left, first of all, is a stochastic model of a landscape that involves two different types. You can call the whites democrats and the black republicans or trees and prairies or whatever. The one on the right is a landscape produced by a deterministic model. And you can see that they're similar. 

Now, another way to measure the similarity is by means of something called the fractal dimension. Now, these objects that I'm showing you, these patterns, are fractals in the sense that you see structure but as you zoom in on finer and finer scales, you continue to see structure. There's structure within structure within structure over many orders of magnitude. And one way to quantify that, there are many ways, is to choose two black points at random in the pattern and ask: what is the probability that they are separated by a distance epsilon, more or less? That they are closer together than some distance epsilon? And you do that for all the black points and compare with all of the other points, and you plot the log of that probability at two points or within a distance epsilon against the log of epsilon, a log/log plot, and you notice that over a large range, over something like almost two decades anyway, it's a straight line. And the slope of the line, this is about 1.6, is the so-called fractal dimension. And the deterministic model has been chosen to give a similar fractal dimension. Now, I mentioned the word fractal, so I have to pause and comment on what's fractal. Some of you probably heard of it, and maybe others haven't. It's hard to talk about this business without talking about fractals because most of the patterns that I've shown you and the ones that occur in nature have this fractal characteristic that they contain miniature images of itself. Fractals are geometric objects that are self-similar. 

Remember in high school you learned about similar triangles? They have the same shape but they're different sizes. If something is self-similar, that means it contains images of itself on ever smaller scales. And a true mathematical fractal would have images all the way down to zero size. Well, here's one example at the top. The word fractal, but if you look carefully the F is made up of the word fractal and the F in that is made up of the fractal and so forth and so on. You can only see a couple of generations because there are limitations of the computer graphics. But if one were to zoom in on this word and expand it, it would perfectly overlay that. So that's the sense in which a fractal is a self-similar geometric object. Here are some other common fractals. This one looks like a snowflake. 

This one looks sort of like some kind of carpet. This one looks like kind of a river basin of some sort but doesn't flow anywhere. This looks a little like a fern leaf. All of these fractals on the left are embedded in a two-dimensional space in the plane of the screen. You can have fractals in three dimensions. Here's one that's called a Menger sponge. It's a funny kind of a sponge. It's got lots of holes in it. In fact, it has no volume at all. It would weigh nothing. 

But it has infinite area. It'd be a great sponge. I suppose it would soak up a lot of water, and it wouldn't cost much to manufacture, I don't suppose. There wouldn't be much material involved anyway. Mathematically you can have fractals in any number of dimensions. You can have a fractal embedded in a hundred dimensional space. Very hard to conceive but they certainly exist. Well, fractals. Now, another characteristic of a fractal is that when you plot something versus something else on a log/log scale, it's a straight line. And such plots are called power laws. 

And I went on the internet and looked for examples of power laws, and in 10 minutes I found four nice examples of power laws. The one on the upper left is something called Zipf's law, and this is words in English text. If you take any book in English, or in fact any language would suffice, and you rank order the words from the one that's used the most common to the least common and you plot the frequency with which that word is used versus its rank, you see something that's approximately a straight line on a log/log scale over several decades. Now, Zipf was a linguist at Harvard and he got very interested in this and he proceeded to do this with lots of other examples, which I'll mention in a moment, but some other common examples are the magnitude of earthquakes. Earthquakes are measured on the Richter scale, which is a measure of the energy, and it is a logarithmic scale. Someone has already decided to take the logarithm, or, sorry, this is the size of the earthquake, the energy, on a logarithmic scale, and this is the probability of occurrence. So there's lots and lots of small earthquakes and a small number of big earthquakes, but they follow a power law over many orders of magnitude. Very interesting. The upper right is the size of power outages. Every now and then the electricity goes down, usually when there's a storm or something, and if you plot the probability that you will lose your power versus the number of people whose power is lost, you find that there is lots of small power outages that influence just a small number of people, a small number of power outages that influence millions of people, and over a decade or more it's roughly a straight line. 

Well, here's another one that I sort of like: internet document accesses. If you plot the number of accesses that a, sorry, yeah the number of accesses that a document gets versus its rank order, you find, over many orders of magnitude, it's very accurately a straight line. Now, there are lots of other examples of power laws. In fact there's so many, after a while you begin to think everything's a power law. So maybe I'd better tell you a few things that are not, or at least one thing. If you plot the distribution of heights of humans, you don't find there's a small number of people that are a hundred feet tall and a large number that are two inches tall. [LAUGHTER] We all cluster about some mean of about six feet or five feet or so. So not everything is a power law. And as I tell the students, hopefully your exam grades are not a power law either. [LAUGHTER] There's not one person who makes an extremely high grade and everyone else gets almost zero. 

That wouldn't be very good. But lots of things are power laws. And that suggests that there may be some underlying connection between all of these different things. That connection has yet to be made so there's a research area here if any of you are interested. Populations of cities. There's a small number of cities with large populations, a lot of cities with small population. You rank order them and plot them as Zipf did with the words, it's a straight line. Sizes of craters on the moon, there's a lot of small ones, small number of big ones. Size of solar flares, same thing. This is one I like. 

The size of computer files. I can predict that if you go home and open up your hard drive and list all the files on it and their size, you will find that there's a small number of files that are really big and a large number that are small. And if you plot them in this way, you'll probably see that they follow a power law. And I don't even need to know what you use your computer for to make that prediction Casualties in wars. There were wars that killed millions, not many fortunately, but there were lots of individual homicides and small skirmishes that killed a small number of people, going on even as we speak. Occurrence of personal names. There's a lot of Smiths and Joneses, but I think I'm the only Sprott in Wisconsin. Number of papers that scientists write. A few of my colleagues write lots of paper, and most of my colleagues don't write very many. The number of citations that a paper receives. 

A few papers get thousands and thousands of citations. Most of the ones I write, I'm lucky if anybody notices. The sales of books, music, almost anything you'd want to choose. Some books sell in the millions, most books don't sell many at all. Individual wealth. This is sort of interesting. Economically, you wonder why there's a small number of people that are billionaires and lots of really poor people. Something has self-organized the economic system so that it's roughly a power law. And it really doesn't matter very much whether you're talking about the American economy or European or Indian or almost anything else. The slope may vary a little bit, but there's always a small number of rich and a large number of poor in any economic system. 

It's almost as if nature has some reason to produce that that we don't know. We don't know why that would be. And there are many, many others. Well, okay, I promised you six models, and I've told you three. The diffusion limited aggregation and the cellular automaton. Now I want to tell you about a model that's near and dear to the hearts of ecologists. Lotka Volterra models describe predator/prey relationships. The way in which the foxes and the rabbits interact and feed on one another. And potentially, this gets a little technical, but I can say it in words if you don't like equations. There's equation at the top that describes the dynamics, but let me just tell you what it means. 

I was interested in the simplest such model that is high dimensional, which means it involves many species, many agents, and I wanted it to exhibit interesting dynamics that is consistent with what you might observe in the real world. So what is the simplest model you could think of? Physicists always think in those terms. When there's something in the real world you want to understand, a good strategy, for physicists at least, is to find the simplest example of that phenomenon you can and then try to understand it in detail. The biologists, by and large, don't approach things this way. They want to put lots of stuff into their models. But you can learn a lot with simple models even though they're toy models and are not realistic for anything. So one simplicity is to imagine that all of these species are interacting, but they're interacting only with two others and they're competing for resources let's say. They're all eating some common food let's say, and when you eat a lot of it, there's less for your neighbor to eat. And the rule is that we're going to put them all around in a ring, making the things one dimensional in space, very unrealistic but it leads to interesting dynamics nonetheless. And there's this interesting rule that you're going to compete with your neighbor on the right but you're not going to compete with the one on your left. 

You're going to compete with the second one on the left. Now, why might you do that? Well, maybe the one on your immediate left is your spouse, and you don't want to compete with him or her. But the model breaks down a little bit because you notice you're not your spouse's spouse. Does that make sense? So anyway, it's kind of an arbitrary rule but it breaks the symmetry, and that symmetry breaking is important for getting the dynamics as it turns out. So you're going to compete. You compete with yourself. The more food you eat, the less you have to eat. Or if you're a whole species of animal, you're competing with the other of your species, but you're also competing with two of the neighbors in this weird way. The interesting thing is the rule is exactly the same all around the ring. Everyone's playing by exactly the same rules. 

Now, your instinct would be to say nothing very interesting can happen. They'll either all die or they'll thrive or they'll settle down into some steady state. But in fact, something very interesting happens. You get dynamics. And here is a plot of the distance around the ring, I've not taken several hundred, I think about 400 around the ring, and this is time going that way, and the black and the white indicates how well you're doing in some sense, how much food you're eating, how healthy you are, any criterion. And you can see this interesting pattern. It looks a little like sand dunes or something. And that connection is maybe more than coincidental because sand dunes also are organized by natural processes that occur in nature. In any case, you can see a degree of order, oops, you can see a degree of order. Things seem to go along a diagonal here as if there's some propagation around the ring that if you wait long enough your turn is going to come again. 

If black is good and white is bad, then if you are here, as you wait, you will go from bad to good to bad to good. But not in a regular periodic manner. In fact, the temporal behavior, if you look at just one position here and follow it in time, in fact is not periodic, it's not stationary, it is, in fact, an example of chaos. So I have to talk a moment about chaos. So one slide about chaos. So, roughly speaking, chaos is the unpredictable behavior of a deterministic system. You'd like to say if you know the set of rules, you could then predict exactly what's going to happen in the future. And indeed you can in the immediate future, but if you try to predict ahead in the long term, you discover that your prediction becomes less and less accurate. This is the so-called butterfly effect. Because a butterfly flapping its wings can completely change the weather two or three weeks from now. 

So small disturbances have big effects. So that's chaos. Chaos has sensitive dependence on initial conditions, the butterfly effect, it produces erratic fluctuations, and it never repeats. No matter how long you look, you will never do exactly the same thing that you previously did. You will always be doing different things. And finally, systems that produce fractal spatial patterns usually produce or exhibit temporal dynamics or temporal chaos. Well, that's what chaos is. Now, I promised six models. The fifth model is going to be a social dynamic model. I stumbled across this in a very interesting way. 

I was at a meeting once and there were some workshops going on and I was a little bored and didn't have anything to do and I asked the workshop leader if I could sit in on the workshop and it was on mathematical modeling. I said, well, that sounds interesting, that's sort of what I do. And he got us in the room, about a dozen of us, and he said, okay, we're going to model a singles bar. And we all sort of looked at each other. It was a meeting of mainly psychologists, and I was amazed at how much expertise there was among the attendees about singles bars. [LAUGHTER] But we proceeded to ask ourselves, well, what does he really mean by this? We finally figured out, well, that's part of the challenge is to figure out how you would model something as complicated as that. And, in the end, we realized that it was really a larger question that was being asked or that we could address that didn't have to do specifically with singles bars but how do people make social connections, in general. And so you begin thinking, well, what is the simplest model you could think of that would represent the way social connections are made in the real world. Well, the rule that we came up with in this meeting was three, three rules. And the first rule is that you tend to befriend friends of your friends. 

Does that make sense? So if you're my friend and she's you friend, I'm likely to be her friend after a while. Makes good sense. The second rule is you tend to mirror others friendliness toward you. So if you're friendly to me, I'm going to be friendly back to you, right? And vice versa. If you don't like me, I don't like you very much either. So those are two of the rules. Now, if you stopped there, we would end up everyone being in a big love fest, or hate fest, I don't know which. We'd sort of go to one extreme or the other. And so we had to put in a third rule to make the dynamics interesting. And the third rule is you have a limited capacity to maintain friendships. 

So if I'm going to be your friend, I have to not spend quite so much time with you two. I'm sorry, but I just don't have enough time. So we all have one unit of friendship that we can spread around any way we want. Presumably, as a result of these rules, it's a deterministic model, and so we're not free to choose how we're going to distribute our friendship. It's determined by what the other people around us are doing all the time. So it's another deterministic model in social dynamics. Well, I can actually, oops, let me see. Let me see if I can run this. Let's see if this works. Yeah, here's the model actually running. 

You can see it starts from sort of an ordered initial condition where nobody is very friendly with anyone, and it ends up in kind of an ordered final state that nevertheless evolves in time. >> What are the --? >> Oh, sorry. A little bit too fast. So there are 50 individuals in this group. And so here are the 50 individuals, and here are the other 50 individuals that they're capable of interacting with. So where things are dark, that means, I don't know which way it is, but let's say the dark is you don't like that person and the white means you like the person a lot. And you see the clustering that has occurred here. There are little cliques that form. There are groups of people that get together and tend to associate with one another, and then there are other groups that maybe they associate with themselves but, in any case, they're not friendly to the rest. And so it's another example where nature self-organizes based on some very simple rules. 

Now, the last example I want to show you is one from pure mathematics, and it goes by this wonderful name of strange attractors. That sounds like a whimsical name, but it's actually a mathematical term. A serious term that the mathematicians use. And when I first learned about these things, I was so enamored with them that I actually wrote a book by the name Strange Attractors, and it has in it 300 some odd examples of these things that I'm about to show you. So although they're purely mathematical, I think you will see when I show you a few examples that they look like things that you can relate to in nature. They remind you of things that you might have seen. So let me just show you one example of a strange attractor and show you the sense in which it is an attractor. So it's going to play around through a loop here, and what I'm doing is starting with a random initial condition where all these points are distributed randomly throughout this two-dimensional space. It's actually a three-dimensional space. And you can see a shadow here, indicating the third dimension. 

It's actually a four-dimensional space, and the color is representing the fourth dimension. And the equation, if you're interested, is in the upper right-hand corner. It's a simple quadratic equation of the kind you learned to solve when you were back in high school. But it is an iterated equation, which means that each step you generate a new position for each point in the pattern and you do this over and over and over again so the individual points move in time and they're moving rather rapidly. Each one of these is one generation. And you can see over about, within a dozen generations, these points have coalesced onto this object that is called a strange attractor. And why is it an attractor? Because it doesn't matter where you start on this picture, you're going to eventually, if you were that point, you're going to jump around and you're going to eventually end up landing on this object. Why is it strange? It's strange because it's a fractal. As you zoom in and look on finer and finer detail, you can see that what looks like lines here are really made up of lines of which themselves are made up of lines and so forth. So this shows you what a strange attractor is mathematically, how simple it is to produce them, and an interesting thing is this equation in the upper right-hand corner, which you can't read, has eight perimeters in it. 

They're coefficients of the eight terms of the equation. Four of them are linear terms and four of them are quadratic terms. And those are like knobs, and you can adjust those eight knobs and every combination of those eight perimeters gives you a different picture. So the same equation with just different adjustment of the coefficients of the eight terms gives you a different strange attractor. And I'm going to just show you a variety of these strange attractors produced by exactly the same equation with different coefficients. So, in a sense, these coefficients, I like to think of them as you've seen these combination locks where you dial in usually about four numbers and then it unlocks. Well, here you have to dial in eight numbers and it unlocks, and what it unlocks you to is one of these pictures. And they're all kind of pretty, right? They're almost artistic. Now, maybe there's nothing in nature that you've seen that's quite like this, but you can see the general form may remind you of certain kinds of plants or turbulent fluids or balls of string or who knows what. But they are sufficiently familiar to you that they're kind of aesthetically appealing. 

And I was intrigued by that idea. Why do we find these things so interesting to look at? It must be because we have trained our minds to relate these to things in nature that we have appreciation for, like clouds or swirling rivers or whatever. So, I proposed the following experiment. I asked about eight mostly graduate students to sit down in front of my computer and look at several hundred of these and evaluate them aesthetically on a scale from one to five according to what appealed to them. And I was interested in whether one could predict in advance which ones they would find appealing. Now, for each of these patterns, there are various things you can calculate. One thing is that fractal dimension that I talked about earlier, each one of these has a single number associated with its fractal dimension, and you can calculate it, the computer can anyway. And so you can ask, well, is there a range of fractal dimensions that people find aesthetically appealing? There are many other things you could calculate. Another thing is something called Lyapunov exponent. It's a little technical but it's a measure of how chaotic it is. 

All chaotic systems have a positive Lyapunov exponent, but the bigger the exponent, the more rapidly you'd lose your ability to predict. And so, just with those two perimeters, I asked whether one could catalog these and find the region in this two-dimensional space where people found the patterns most aesthetically appealing. And the answer is yes. Here on the horizontal scale is the fractal dimension, and here on the vertical scale is the Lyapunov exponent. So this is a measure of how fractal it is, and this is a measure of how chaotic it is. Two related ideas. And the black regions are the ones that people liked a lot. The white regions are ones they didn't like so much. So there was a range of fractal dimension between about 1.2 and 1.4 or so which people liked. And they seemed to like things with a small Lyapunov exponent, curiously, even though the Lyapunov exponent is a dynamic measure and the pictures were perfectly static. 

A little bit of a mystery there. But anyway, I thought it was interesting that these ideas of self-organization apply throughout the sciences and throughout nature. I've shown you examples from astronomy, from ecology, from economics, from social psychology, from pure mathematics, and now from art as well. Now, the idea that one can quantify aesthetics is challenging or threatening to many artists. And, in fact, many artists would say those patterns aren't real art because a computer did it. Well, there's kind of a Turing test you could apply. If you saw those patterns and didn't know how they were produced, would you be able to know that a computer did it rather than a human? Perhaps, perhaps not. In any case, the idea that things self-organize is familiar to us all, and there are examples throughout nature of this. So let me just conclude by summarizing what I hope you'll take away from this talk. The first thing is that there are many simple models. 

I've shown you six, but one could generate dozens more very easily. Simple models exhibit self-organization which I will define as the spontaneous development of complex structure. Complex here means not random. There's a degree of order, but there's also a degree of disorder associated with these structures. Models with simplistic rules can have highly, or, sorry, with symmetric rules can have highly asymmetric behavior. This is an example of what's called symmetry breaking. Many of the examples I showed you, everyone is playing by exactly the same rules, so you would expect the pattern to somehow evolve into something that has no spatial variation. But in all of these cases, that was not the case. They evolved into something with intricate spatial structure. Thirdly, I didn't talk very much about this, but some effects may not have easily identifiable causes. 

One definition of science is that it's the study of cause and effect relationships, and that's what we were all trained to do in our training as scientists. Sometimes we observe the effect and we're looking for the cause. Sometimes we do something and we want to know what effect it will have. But, in a sense that's what science is thought to be about. These complex systems usually do not have simple cause/effect relationships because they are replete with feedback loops, and, as a consequence, every effect causes a change everywhere which in turn becomes a cause for every effect. So there's this idea of iterating things over and over with everything changing generation to generation, changing all at once. Now, a lot of people object to this whole idea because they say, didn't I learn about the second law of thermodynamics way back in my first physics course and there was a law of physics that says systems always go from order to disorder. And yet you've shown me six examples of things that are doing exactly the opposite, going from disorder to order. Doesn't that violate the second law of thermodynamics? Well, this is where some people appeal to some kind of super intelligence saying, no, there must be someone figuring out what it's supposed to do. But I've shown you, at least in these particular models, there was no great intelligence involved. 

Things were just obeying simple rules and yet the organization occurred. The solution to this is that the second law of thermodynamics is devised for systems which are in or very near equilibrium. Which means that they have sort of achieved a balance with their surroundings. In all of the systems I've shown you, they are far from equilibrium. Now, how does something get out of equilibrium? Well, you have some throughput of energy or resources or something through the system that's coming in by some process and then it's going out by some other process, and this keeps the system pulled off the state of equilibrium. Now, that's a little technical so let me just give you an example. An ecological system, you'd say how can we have all this complicated ecology? Where is all of that order coming from? Well, it's coming from the sun, ultimately. If you turn the sun off, I can guarantee you that all that complex ecology would disappear very quickly. Everything would die. It would indeed reach a state of great disorder. 

And it can achieve the order only because it's dissipating the order that the sun has by virtue of it being so much hotter than the rest of the things that it's supplying energy to. So it's this difference in temperature. And that, in fact, is why in the big bang, even though 400,000 years after the big bang things were highly uniform, that it nevertheless could self-organize because it was extremely hot. It was very much out of equilibrium. An equilibrium system wants to cool down and become uniform temperature everywhere. Indeed someday the stars will burn out and maybe we will go to a state of great disorder, but for the moment at least, lots of self-organization is happening on the natural scale. So I promised one comment at the end about religion. If any of you are threatened by anything I said, and there was a time when people, they used to call themselves creationists but then they began to say they believed in intelligent design, said that when we look around at all of the complexity in the world, like you and me, there had to be a creator, an intelligence behind it. Well, I hope nothing I've said has dissuaded you if that's your view because I don't have a counterargument to that. But what I would like to say is that if there is intelligent design in nature, it is probably happening at a much lower level than is generally believed or is generally considered, that things are able to self-organize and develop extreme complexity out of pure randomness because there are simple rules that are being applied over and over and over. 

Now, if the rules were produced by some intelligence, that would not be inconsistent with anything I've said. So, with that, I will just conclude by giving you a reference. If you want to see this talk, it's on the web. That's the URL. I have a book on chaos that talks about a lot of this stuff. If you want to contact me, there's my email address. Thank you so much. [APPLAUSE]