National Collegiate Honors Council, San Francisco, 31 October, 1996


I feel very honored by this invitation, and I like the introduction also. I'm now going to take it seriously. I think the important thing here is the subject. You have invited not me but the subject of "Chaos and the Millenium." And although I can't personally do it complete justice, I do think that it's a very important subject, and it's very appropriate for this time, and it has very important indications for education in general and for honors programs in particular. And I have learned, since receiving Ms. Rupert's invitation, that honors programs actually provide the opportunity I have for a long time searched for and not found--to teach the kind of information I think that is important in universities. So I want to try to address this subject in the perspective that it's something for you to do, something for us to do, in the reformation of education on all its levels and the attempt to address the future and successfully create a future society which has sustainable possibilities.

So I want to proceed like this. I would like to give a short as possible lecture, presentation, followed by as long as allowed question and answer session. And there's a microphone here for your questions, so when we get to that point, you may just run up to the microphone and ask and everybody will be able to hear. And I'm counting on you for this. Some are obviously not so good at this, but I have high expectations for you. You might help me out with this, because that is the best mode for our communication. We have here a subject which for a normal university audience would be very arcane, but I think that I am feeling very much at home here, and somehow, I don't know why, the timing is very appropriate. 

So, in this short lecture I want to discuss three things: The first is Chaos--what do I mean by that; and secondly, the Millenium--what do we mean by that; and finally, but briefly and most importantly, Chaos and the Millenium--how it goes together in an essential way. 

Chaos, you know, is the chaos that you know, but for me it's Chaos Theory, which is a brand of mathematics. So here we are in a formal reductionist hierarchical university structure. Everybody experiences chaos, but I'm presenting it as a branch of mathematics. So the question arises, and has always arisen, as to whether or not there is a connection between the mathematical model of chaos and the chaos of everyday life or whatever is the (organizing tension?) of that word. And for a long time I ** this connection, but now I feel it justified. 

So mathematics has those branches or further divisions which are standard in the history books of arithmetic, geometry, dynamics, and analysis, with geometry to follow * and so on. All these branches are old, more or less old. Algebra, Analysis, Geometry--these are ancient subjects, coming to us from ancient city-states like Shumer Babylon, Hindis Liber and Ancient Egypt. Dynamics is newer, only three or four hundred years old, but still it's called classical, Classical Analysis. If there were to be a new branch of mathematics, it would be a tremendous novelty and pose a challenge to the orthodoxy of mathematics at universities. Because for a long time there has not been a new branch of mathematics. Now there is. Chaos Theory and Fractal Geometry are new branches of mathematics. 

So my first subject then, is, Chaos Theory has a brand new branch of mathematics and one which has evolved particularly into a mass phenomenon, a major social transformation called the Chaos Revolution, primarily, I think, because of computer graphics and the computer revolution, and the possibility for the first time to make the arcane and secret images of mathematics visible to all and everyone on the cover of magazines like Scientific American, the cover of books, and so on. And people are seeing computer graphic images of mathematics for the first time, they have seen what mathematicians have always seen in their imaginations, and they have found that they're beautiful. And therefore there's been self-organization and augmentation of mathematics because of the long-lost connection between what mathematicians secretly do and what everybody else does, including business and making political policy and so on. 

So in order to continue, I wanted to introduce in as short as possible time the smallest number of essential concepts from Dynamical Systems Theory, also known as Chaos Theory, so that we can use them in conversation...[turning on projector]

There was at the University of California at Santa Cruz a visual math project in which we tried to use computer graphics to help teach mathematics in the elementary courses, and after funding for this project ran out, I repackaged all our work as the drawings in this book, now out of print, but soon to appear on my website, which I imagine to be the alternative educational system of dreams. 
So the minimum number of concepts I want you to acquire here, three kinds of attractor and three kinds of bifrication, and along the way, hopefully--just the vaguest idea is all we need, what's an attractor, what's a bifrication. 

The kind of dynamical systems that we are talking about here, under the name Chaos Theory, it starts with this--you see this rectangle here, this is the never-explained secret of * which is at the basis of everything and goes back to Geometry. This is an imaginary mathematical model for a system, some system. Let's say you have a restaurant in Chinatown. The vegetables and rice come in the back door, and the satisfied customers go out the front door leaving money behind. Now you count up the vegetables and you count up the money left behind, and then you make a model for this, and the geometrical model is called the (state space?). That never is simple, but in this case is the garden variety ** (field in the plane) through this rectangle, which has the following interpretation: every point of that rectangle is supposed to be the representative, the avatar of a state of that kind of restaurant. Now it's not a very good model, and it's not very rich and for example the slavers of the meals are not there. So in this reduced model, anyway, the point is that each and every point in this geometrical object is the representative of a real state of some other system that we are pretending to model in this very very simplistic way. 

Now enter Dynamics, which begins around the year 1600, the very same time that Jodorno Bruno was burned at the stake for thinking about these things. The system changes, like every minute it has a slightly different state. so its representative is wandering around in this state. The representative of the system in this living evolution(ness) is a moving point in that space that draws a curve, which is called a trajectory. And here's where the really simple-minded mathematicians put their restrictions. It is imagined that at every point in this space there is such vector which gives the running instruction where that trajectory is going to go next. That means that in case you were here, you must go in that direction, if you are the trajectory of the model--not the trajectory of the restaurant, of course; the restaurant can do anything it wants--but the model is extremely restricted, okay, and this ** field means that there's a fixed running instruction at every point on the playing field. 

Now, the idea of a tractory(?) is, suppose we have such model and we are interested in the long-run behavior of the system. That means we only want to know, twenty years from now, if there will be a fortune to pass down to the next generation or not. So that means that in order to answer to this question, What is the long-run expectation of the system in the model?, that we have to start from one point or another point of many points, and follow the instructions, drawing the trajectory of the system for a long long time. And when you do that, you come to what is called an attractor, and it's a different kind, and here's the simplest kind. Here's an example of a trajectory which gets closer and closer to a final state, which is a point, and when it arrives in the neighborhood of that point then it essentially doesn't move anymore. So this is a mathematical model for death as it were. That would be--I don't want to say anything political, but...So there's three kinds of attractor and that's one kind, and that's called a point attractor, otherwise known as a static attractor, and it's a good model for death. So without--the fact is we're running short of time. Let's look at a model for life. 

Here's one where the long-term behavior gets closer and closer to this cycle, this loop. And when this trajectory--no matter where you start--if you start inside you end up going around the loop, if you start from any point outside you will end up going around this loop, and the implication is that after the so-called transient dies away and then you arrive near that attractor, which is called a periodic attractor, then you are repeating the same cycle of states over and over again, and each repetition taking exactly the same time. So that's a model for oscillation, and as many living systems oscillate we can say that's a model for life, or somewhere between life and death anyway. It's kind of a very simple life. A periodic attractor, also called a * cycle. 

And a third sort is called a chaotic attractor, and amazingly, it was discovered recently, and it came really unexpectedly. It was a violation of expectations, and therefore I think people would have been able to deny the existence of this model for chaotic behavior if it weren't for computer graphics, which made it mandatory to accept the existence of this thing. So this model is actually in the three-dimensional space place. The lowest dimension in which this occurs is three dimensions. So you have to have kind of sophisticated computer graphics to even visualize it. But starting from any point in a two-dimensional * around here, the trajectory gets closer and closer to this object, and it releases it in the following way. It goes around this as sort of a point of a wing, which is then bent up, stretched sideways, and the upper lip folded over again, so that outside edge comes down on the inside edge, so that they're actually down here as you may not have suspected, two layers not just one. Now you can follow the two layers around and fold them over and then here's two layers and here's two layers and after the folding then they're four layers. So if you do this again and again, eventually you'd build up an infinite number of layers. But they are pressed flat into a very thin--well, thin like filo dough or croissants, baklava, something like that--which is an example of a fractal object, and which indicates just a little bit the overlap between these two new branches of mathematics, Chaos Theory and Fractal Geometry. Here are some drawings--my co-author here, the artist Chris Shaw, has done a particular good job, I think, in indicating in ** drawings the infinite fractal structure of this object. For example here, what we have done is take a slice of the thing, slice through that filo dough to look at the cross-section, bring it up here, and then you see this--well, it's a typical fractal. There is a lot of layers, but there are big spaces between some layers, and in each space there's some layers, there's slightly smaller spaces between them, and--we couldn't draw them all.

Okay, there's three kinds of attractors, and they are modeled for the long-run behavior of an idealized mathematical model for practically any kind of system which is a dynamical system subject to this very rigid restriction that the rules of evolution don't change in time, so-called autonomous dynamical system. No suppose the rules do change in time. Then these attractors change. One kind of attractor could change into another kind. An attractor could move. Because this picture in which the attractors are indicated in a state space, the picture depends on the rules. So if you change the rules the picture changes, and what was found--in two ways, by the observation and also in computer simulation, the observation of mathematical models--what was discovered is that the ways the attractors change in a dynamical system when you change the rules is waves are extremely [recording ends here]

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