 at the University of Bremen. Since 1985, he has also been Professor of Mathematics at the University of California at Santa Cruz. He is a co-founder of the Institute for Dynamical Systems at Bremen. Professor Pyken is widely known for his work on the generation of fractal images. Two recent books include the Beauty of Fractals and the Science of Fractal Images. Yesterday evening, Professor Pyken was asked whether the fractal images that are on display in our campus art gallery are truly art. Whether art or not, one cannot deny the exquisite complexity and mathematical beauty of these fractal images. Please join me in welcoming Dr. Heinz Otto Pyken who will speak to us on the topic, The Beauty of Fractals. Thank you, Michael. I'm glad that we kept it short so that I have a bit more time to run my experiments. Before I say anything about factors, I want to say this. I think anybody who has ever organized a meeting with SARTI participants know how much work is involved in doing that. And some of us have organized meetings with a hundred participants and they know it's a quantum leap to go from SARTI to a hundred. But to imagine that one is able to organize a meeting smoothly and such successfully which has more than 3,000 participants is almost a miracle. And I think that's a wonderful point in time for me to extend our gratitude and our admiration to all who were involved in this marvelous event. And I think I would like to have to speak in here if we want to. Thank you very much. I am just the tip of the iceberg and on behalf of the many people who made this beautiful conference possible. Thank you. Thank you, John. All right, I would like to start with a few slides. Pat, can I have the first slide, please? My first slide is a cover image from Scientific American of August 85. And I thought I'd start with that because this image by itself created something which never happened before in the history of mathematics. This, as you all recognize, is a blow-up of the Mandelbrot set, which was discovered by Benoam Mandelbrot around 1980. But after the publication of this image and the code to generate the images very rapidly, hundreds of thousands of young people around the world started to implement the code and generate their own images. And in some sense, one might say, soon the Mandelbrot set became the brightest star here from science and asked ourselves, what does that tell us, in a lie? Can we make use of that in seeing the bar? That here is an attempt that you shouldn't take to see it, but let me expand on it anyway. What you see here is the city hall of the 16th century. And as you see, it's a pretty beautiful building. You might ask yourself, why is it seen as beautiful? Let me show you. Let's just walk closer to it. And as you see, getting closer, we see new and complicated and nice stuff. And closer, we see more detail, more and more structure appears. And I honestly believe that many people do, that that is one of the reasons that we perceive such pictures, such buildings as interesting, maybe beautiful. Now let's switch to the new town hall of Bremen, which is next to the old one, as you can see. And let's do the same experiment. Let's just go closer for a little bit. Well, how interesting. And as we get closer, even more interesting. And guess what we see if we go even closer? And closer even? So I hope you don't take this to serious, but I think to some extent there's some tools in that. The repetition of structure as you go closer to an object makes it interesting and maybe beautiful, if it's done right. So let's maybe stop at this point, this is light, and start to prepare to take what I want to do with science in the invisible. There's two kinds of factors, if you take a very, one kind which you can see as you walk through nature, but there's another kind which probably is much more important even, and that is a kind of factor which you see only through research, through scientific work, through experiments. And here's a very nice and simple example which displays that. Please start the tape. On a thread exposed to the force of gravity and the attraction of three magnets. Up to any starting position and then released, it will ultimately come to a stop over one of the magnets. But over which one? That depends solely on the starting position. Each magnet will attract any ball that is released in its immediate neighborhood, but each will also attract some balls whose starting position is farther away. The situation is complex and extremely difficult to grasp within the framework of the experiment. Here computer simulation can be of help. We mark the ball's starting point with the color of the magnet over which it comes to rest. To obtain a systematic overview, we do this for a series of points scattered uniformly over the entire plane. The colored squares represent only their central points. It is possible to refine this structure by using a denser grid of sampling points or an even denser grid, an even denser one still. The colors are stirred together into a highly complex structure. How should we understand this picture? In some areas it is quite clear. All balls that are started within a blue area land over the blue magnet. From here the ball lands on the red magnet. From here starting from only a slightly different position on the yellow one. And from a position in between on the blue one. The magnets seem to compete for the balls down to the smallest detail. Any prognosis as to the behavior of the balls in this area is nearly hopeless. This is chaos, pure and unadulterated. So far for this experiment in which I think you have seen very nicely there are fractals which we cannot see by just walking around. Those are fractals which we will see through research, through experiments, through careful investigations even in classical systems as this one. But now let me go on and come to the second important ingredient to understand fractals. And that is the notion, the idea of a feedback loop. What I mean is something very simple in principle. Some kind of a black box if you will which has an input and an output. And whatever comes out of the machine is fed back into the black box right away. And you should imagine that the system is running in time and as it runs something evolves. And of course what we want to see evolving is structure, pattern, fractals. And typically such a system will depend on an outside control. You should think of a gear shift which we can put in second or third or reverse. And in doing so the system will behave differently and the task of mathematics is to describe what happens for different settings at least in the long run. What happens if you watch the system evolve after a long time. So that is the second ingredient, the first one self similarity, the second feedback loops. And now let me continue with a first experiment. A physical experiment, an experiment which is known to some extent for as long as we know video cameras. But to another extent it's not known and that's interesting and peculiar. And I should say what I will show you now was suggested to me by my friend Ralph Abraham from the University of California in Santa Cruz. The principle setup is very simple indeed. All you need is a TV camera and a monitor. And what you do is you have the camera looking at the monitor. And whatever the camera will see will be fed back into the monitor. So that's the closed feedback loop. Now I should tell you that there has to be a little bit of a trick otherwise you won't see much. And that trick is almost visible as you can see or not I will tell you. The camera here is put upside down. And of course that creates something funny. It creates basically a rotation of the images which appear in the system. And if you cannot do that because usually cameras like to sit on a tripod and don't want to be turned over, you can equally well take the monitor and turn the monitor upside down. That's exactly what we have done in our experiment here. Let me carefully walk over here. You may or may not see that we have a monitor and it's really sitting upside down. Let me show you the cable to it. It's sticking out on the top. And here's our camera. And everything is set and nothing happens. And that's clear because there's no picture on the monitor and consequently nothing is seen by the camera and consequently nothing is happening. So let's now start to fire up the system. We need some initial event. And here is my initial event which I want to take. Do you see it? I will take this laser pointer and I will shoot at the monitor and you see oh pet, did we switch on or not? We didn't. Can we have the video please? I didn't tell you. I'm sorry. Is it on now? I guess so. Okay, so again I will start to fire up with my laser pointer and now you should see. No, pet. I'm so sorry. What happened? We have a remote control for the video switcher because we have to switch between all kinds of equipment and apparently the video switcher didn't work. Did it work now? Okay, so let's start again. The last time. Every experiment needs three trials. Here we go. I'm sure by now you question what you see. I'm sure that you believe that pet was going to the back to switch on a VCR, right? In fact, if I would see such a picture and somebody would tell me that's simply generated by such a simple feedback loop, I would say no, impossible. But how would you explain this? Do you see my hand? Did you see that? In other words, it's live. Let me fire up again before it dies. This is a live pattern. The pattern is generated simply by that feedback loop. And once you have such a pattern, you can start playing with the outside ear shift. And here's one, you can take the camera in your hands and you can start to rotate it about its axis. And if you do so, hopefully the picture doesn't die soon. Do you still see something? Like let me open the iris a little bit. Well, live demonstrations as you see are difficult. I didn't see exactly what I wanted to show you. What I wanted to show you is a dramatic change in the picture as we, but now we see something different, I hope, as we touch the outside controls. In other words, one setting of a feedback loop is able to generate all kinds of complicated structures by just changing the outside controls a little bit. Right now I'm playing with the zoom capability of this camera. And of course, you also would like to see maybe classical feedback. The monitor inside the monitor inside the monitor. Let me do that also. That's what you see here. You see that? And that's obtained simply by moving out by the zoom capability of the camera. So that's video feedback and the demonstration. I think very convincingly that very simple feedback loops are able to generate, if you do it right, right now I'm not getting the point back. If you do it right, are able to generate very complicated patterns. Here we go. Okay, so much for the video feedback. Thank you, Pat. Let me now continue to try another little accident. I'm hooked here with my cable. Let me now try to put a mathematical hand on what we have just seen. In doing that, I want to continue with another experiment, which has become very popular in mathematics as the chaos game. Incidentally, this particular experiment has been around in mathematics for some sort of years, but it was recently made popular by Michael Brownsley. The chaos game is a true game. And as a true game, what we need is a die. Here's the die. Can you see the die? And we need some points in the plane, which we have colored. And since you can see the colors very well, let me just use numbers. We have numbered them, let's say one, two, and three. And now we number the faces of the die also, one, two, and three. And the opposite faces are numbered the same, one, two, and three. So that is a setup of the game. And now we start playing the game. To play the game means we have to pick some initial point, which is our game point. It can be somewhere, anywhere in the plane. And now we roll the die. And of course, rolling the die just means that we pick randomly one of the three events, one, two, or three. So let's roll the die and let's suppose that I do this and I pick randomly a two. What will I do? Well, I will mark from this game point, I will walk towards two and put a new game point exactly in the middle. That's the first run of the game. I roll the die again. This time maybe I roll a three. From the actual game point, I now walk towards three and mark a new game point exactly in the middle. And maybe I roll another three and then a new game point exactly in the middle. And maybe finally I roll a one. Then from the current game point, I will walk towards one and put a new game point exactly in the middle. And as you see, I've now reached the interior of this triangle of the three vertices one, two, and three. And once I'm inside the triangle, I will remain inside for all times if I continue to play the game. For example, if I now would hit three again and maybe then two and then maybe two again and maybe one again, I would create points after points inside the triangle. At this point, you will say, what a boring game. What a boring game. Indeed, it looks awfully boring, but as you can imagine, we bring this game into this presentation because something peculiar happens much like the feedback with the TV camera. Something unexpected to most will happen. So what will happen? Well, we first have to ask a question. The question is, what happens if we play the game, let's say, a couple of thousand times and always mark the current game point? Or more specifically, what is the structure which will emerge if we do so? Well, you would say still very boring because that's clear. Since we are using randomness in creating the game points, the structure which will appear will be a totally random structure, some cloud of points, totally boring and uninteresting. Well, let's see, let's implement this little game which is just a couple of lines of code on a PC and see what happens. Can we have the computer on please? At this point, I will start the game, hit a button and we will see execution of a game in such a fast way that a couple of thousand game points will be generated very quickly. Now I go. And as you see, a structure emerges, yes, but certainly not a random structure, very orderly structure, a structure known to you as the Sopinski Gasket, a structure of pure and perfect self-similarity. But remember, this structure is driven by a random generator. Well, at this point, you would say, I didn't see that in the picture, in the picture which you started on the overhead, those points which you generated initially are not on the Sopinski Triangle at all. You're quite right. You have to forget, let's say, the first 10 or 20 or 50 because soon the game points will settle down on the final picture, which you can almost see here if I fit this picture into the game. These points which were later points are already on the Gasket and remember, I did them by hand. So in other words, you don't believe that, right? In other words, randomness combined with very simplistic rules is able to generate complicated patterns and structures, very similar to what we saw in the feedback game. My next step will be to explain what you see in this situation to some extent. But before I do that, let me just for later reference remark that it doesn't really matter how you play the game. If you take different dice or on a computer, various random number generators, you will always get the same final picture. The way you create the picture step by step, of course, is changing, but the final picture, the net result, the attractor, as we say, is the same. And that's a marvelous property which we try to understand a bit deeper in some 10 minutes from now. Before we do that, let's take a different point of view which is even simpler. And that is by talking about what I call the chaos copying machine. The chaos copying machine is essentially an ordinary copying machine, except that when you open the front door, you will notice that it has three lens systems instead of one. And each of the lens systems is essentially an ordinary lens system, which means it's able to reduce a picture. In other words, if you take a picture, put it on the copying machine and push button, out you get three reductions and those are put on the copy. That's essentially the description of the machine. And if you find this too realistic and like something more abstract, here is the blueprint of this copying machine. Some initial image, which for convenience is a rectangle here, is reduced three times by the same factor of one half and the three reductions are positioned somehow, as you see in the blueprint. Incidentally, the reduction factor one half, of course, is exactly go always half way in the chaos game. So that's the blueprint of our machine and now we want to switch to the computer again because I want to show you how this copying machine works in a computer simulation. So let me bring up a test image, a rectangle for convenience. And as you saw in the blueprint, we copy once and out we get three smaller copies, which I color red, green and blue according to the lenses participating. And now what will we get if we repeat? Feedback. The same as over there with the video camera experiment, we will get a new image, which will be just three times three smaller images and here we see. And then again, 27 and then 81 and then 243 and you see literally how repeating this copying machine again and again, the feedback loop, the dynamical system provides a final picture, at least close to it, which is in picture, the Soplinski triangle. And now you might ask maybe to do this, it is important to pick the right kind of initial picture. Maybe the measures, the dimensions of this initial rectangle matter, the proportions of the rectangle, for example. Or you could ask yourself, what happens if I take an arbitrary image and put it on this copying machine? What will I get? Well, let me show you. Let's take one, which I have prepared, which is not quite arbitrary but good enough and let's see what we get. Well, I think you understand quite clearly three smaller copies and then nine even smaller copies and then 27 and 81 and 243. And at this point, you hardly can see the letters USA, but they're still there. However, have become so small that the figure of all these small letters put together already looks like a Soplinski triangle. In other words, what we have just seen in this demonstration is, it does not matter how you start. No matter which picture you take, the final picture, which this feedback loop will generate, is always the same. And in fact, that's very similar to what we saw over here. If we want to switch back to this just for a second pet, can we do that real quick? Let me try to start it. And of course, now it won't start. Well, here we go with our image and that's some kind of a final image. And as I go ahead and take my finger into the picture and disturb it to bed, now it's gone. But essentially, what I wanted to show you is if I disturb the picture, in other words, change the initial picture where it starts from, the same final picture essentially is coming out again. So that's a very nice property of such feedback systems. They create always the same picture. In other words, this very special blueprint which I showed you is the blueprint description of the complicated Sobpinski triangle. Now, this morning when I saw this nice logo over here, I thought maybe I'd try this logo too. Let me try this pet. Can we go back to the video real quick? So let me try to find this. I hope I find it. Here it is. So we put this on our copying machine and what will we get? Well, I think by now we understand, right? Again, the Sopinski triangle, however, this time it's upside down. That's all. So you see again and again in those experiments, the complicated structure of a Sopinski triangle is completely described, fully described by such a simple blueprint of three reduction lens systems, their reduction factor and their position. So let's now go ahead and look at another blueprint which has a historic annotation. This time we have four reduction lens systems. Each time the reduction lens reduces by a factor of three. And let me prepare my computer experiment real quick. I guess most of you know quite well what I'm going to get. Here is my initial picture. And as we copy, we get four copies. And then we get 16 and then 64 and then 256 and so on. And you see very rapidly another familiar structure, the mergers, the famous Van Gogh snowflake curve, which looks like at least a bit like a natural object, a snowflake. But remember, those structures were known at the turn of the century or a bit later. And they were known in mathematics as mathematical monsters. They were not designed or invented or discussed by the attempt to describe nature or natural phenomena. They had some intrinsic mathematical meaning which made them called mathematical monsters. And there were a couple of others like the Cantor set and the Manger Sponge, not too many, and that was it. And now let's make a jump into the 80s. To make a jump into the 80s means I will now show you a copying machine which is built by the same principle of several different reduction lens systems. However, this time we allow the positioning and the reduction factors of the lens systems to be quite different from each other. In other words, you might think of this as taking such a copying machine and really move it out of order if you like. Here is the initial image, a large rectangle which is reduced by one lens system to the red part which is a very minor reduction as you can see. And then there are two lens systems, the green and the blue one which reduced by some factor of five or six. And there's a false one which you almost can't see because it's a very skinny rectangle, rectangle which is of a new kind because it means that horizontally the reduction is much more than it is vertically. That's also allowed, why not? So that's the system which we want to play now. Let me start my computer. Here we go. Here we have our initial image and now we copy. And what you see is exactly the blueprint which I showed you earlier and you should watch here is the little skinny rectangle right there. And now we copy again. 16, 64, 256. 1024 and you see a structure emerges which has more and more leaves it seems. But also you see it will be quite difficult to see the final picture in this case. And why is that so? Well, remember the reduction lens systems involved in this case are of a different nature. For example, this red one reduces only so little. In other words, to see the final image or to get away from the arbitrary initial image which here was a rectangle, you would have to do many iterations, many feedback cycles. And in each such cycle, of course, you have to compute and display four times as many rectangles in this case. So in general, you have to display and compute four to the power K. Now there's a very simple estimate. How many iterations do you have to do? And the answer is some 40, 4O iterations which sounds like nothing for a computer. But remember, 40 iterations means you would have to compute and display four to the 40 rectangles which is about 10 to the 30 rectangles. And now you might ask how long does that take? Well, you might say here in Minneapolis, let's take a supercomputer. Well, if you would take a supercomputer to compute and display 10 to the 30 rectangles, you would need some billion years. In other words, it sounds like here's a nice mathematical idea, however vast it is because you can never see the final result. Of course, there are some tricks, maybe you say, whatever is so small, I don't care to compute anymore, but that's already a complication. So at this point, we switch back. We switch back to the chaos game. We switch back to the idea to take a die, roll the die, and pick one of the four possible length systems and apply it. In other words, what we now do is we take some initial point like this. We roll the die and we apply to that point a length reduction. That gives a new point. Maybe like this. We roll the die again and so on. So in other words, we play the chaos game with this particular set of reduction length systems. So let's do that here. Let me switch to black so that we can see better and here I hit the button. And remember, if we would do the ordinary algorithm, we would need some billion years on a cray. So let's see how far we get with the chaos game on a PC. I start the game now and go. And you almost see the final picture instantly. So here we have cut down the computer time from a billion years to a couple of seconds. And why can we do that? Well, because there's some beautiful mathematics behind, which we can understand. And I will at least touch this part of mathematics. By the way, this image is now known as the Barnsley Thorn, Michael Barnsley, a researcher and mathematician from Atlanta has some wonderful ideas to a certain extent already carried into engineering applications to use this idea of reduction length systems to code images. Maybe I should just make a remark on that. See, this phone which we have up here is just obtained by applying the chaos game to these four reduction length systems. And then we obtain this complicated picture. In other words, you could view this as the blueprint of this picture. That's the idea which Barnsley is propagating. Now, if you ask yourself how much information do you need to code the blueprint? That's very little. You have four reduction length systems. Each of them is in linear mapping and FI linear mapping given by six numbers. Four times six is 24. In other words, 24 numbers code the blueprint. In other words, 24 numbers, together with the chaos game algorithm, code this image. So that's some interesting and cute idea, but I can't really tell you how far Barnsley has gotten with it because he keeps it secret. If you want to, we can talk about it a little bit, but not too much because as I say, it's kept secret. So let's now touch the mathematical background of the chaos game a little bit. To do that, I should extend the idea of the chaos game just a bit. Remember, I had the idea to take a die and roll the die and choose the reduction length systems accordingly. What does it mean to take a die? Well, a die has six faces and usually the perfect die then would give you equal probability for each of the faces. And that's exactly what we should do in the first attempt. In other words, let's give equal probability to each of the length systems of these four and run the chaos game. Let's do that. I have to open another piece of software now because that's a bit more extended. Here are the setups and now I hope I can go and see what this does for us. Here I go. And as you see, nothing happens, that's all for it. Wow, wow, wow, wow, what happened with this piece? You see, that takes a moment. I have to now change the probabilities real quick for you and I will do that because I really want to show you this effect. I will change this probability. Maybe you can read that there to a quarter. And I will change this probability to a quarter. Uh-oh, I'm running into trouble here. Trouble, trouble, trouble. But maybe I can manage, we will see. Uh-oh, doesn't like that. These are live demonstrations. Here's another one, four of them which I have to change no more. I don't like this, 25. So now I have set the probabilities to, see there's one thing which will never work because there's a decimal point missing. Now we have it. So now I have set the probabilities to a quarter and now let me try to see, what did I? Did I do something wrong? Oh, yeah, I have to separate those lines if I can. Thank you for seeing that. Ah, I hope it's correct now, let's try. So here we have changed the probabilities to a quarter and now let's run. Here we go. And now please believe what you see. This is not because I was the king in the wrong numbers. I really keyed in a quarter as a probability for each of the land systems and you see something is growing but you also see that something is not growing very fast. And now you could ask yourself, how long will it take until it will emerge into the final picture which we know already is a complete phone? And the answer is, some billion years. In other words, what did we do originally in this experiment? We did not assign equal probability to the four land systems. We rather assigned very specific probabilities. And to give you a rule of thumb, we assigned a very high probability to the reduction land system which reduces only a little because we want that artifact to disappear soon and we attach very low probabilities to the reduction land systems which reduce a lot. In other words, you see playing with probabilities in connection with the chaos game allows you to speed up the generation process tremendously but mathematically much more is attached to that. The structure behind here in mathematical terms is measures, more precisely invariant measures and those can be used to code the image even in color. And I just wanted to touch on that perspective to show you something very simple as an idea here very quickly emerges into some very nice mathematical problems. So far, the pictures which we have generated are all self-similar, not strictly self-similar because the stem of this fern does not look like a small fern, it's just a stem. But surely the leaves of this fern look exactly like the entire fern. And that's simply because we copy by the same machine over and over again. And I should say if this particular idea which I'm just evolving now would be limited to pictures of a self-similarity nature as this it would not be worth much. Why? Well, simply if you look at ferns for example you would easily find ferns which are different, ferns which do not have major leaves of this kind, of what kind? Well, there are ferns for example which have a different self-similarity structure. The major leaves branch off with some offset but the little leaves on the major leaves branch off opposite to each other. And if that happens of course the little leaves are not exactly a reduction of the entire fern. In other words, self-similarity is broken. And the question is, is it possible to extend this approach so that breaking of self-similarity can be manufactured in a designer fashion? And the answer is yes it can. And I just want to touch on the idea because it's so beautiful and simple at the same time. The essential idea is that you take two copying machines with their own individual land systems. One which creates the entire fern and if you would just run this machine it would give you a self-similar fern. Another one which creates a major leaf. And this one for example creates a major leaf on which the little leaves are standing opposite to each other. That can easily be done by just adjusting the reduction land systems properly. So now you have two independent reduction land system copying machines and each of them produces something independently. And now the big step is really a little step and that is you take these two independent systems, decoupled systems mathematically speaking and you couple them. You couple them in such a way that now the major leaves are copied into the large picture. So in other words you create a networking of copying machines. Now maybe there's time to show you that in a computer demonstration. I'm afraid it's not something as wrong as this little thing. So I guess not to lose time, I will just leave it at that and summarize before I go to the conclusion. What we have seen so far is a very simple approach of using the idea of feedback of copying machines even up to networking of different copying machines to generate complicated structure using very simple maps, very simple transformations. And in essence these transformations are all linear transformations. And as we learned from Mitch Feigenbaum yesterday very nicely, linear just means in this case you do something to pictures here and it does not depend where you put the picture. If you put it there and apply the reduction land system it will be reduced by some amount and it will be reduced by the same amount if you put it there. That's the effect of linearity here. So that is the first and major part and now I would like to get to the conclusion. In the conclusion I would like to tell you what we have seen so far is at best only one dialect of a whole family of languages to create structure out of those principles. The dialect which we have seen so far is the linear dialect because the transformations the reduction land systems are linear in nature. So let's now go to just another one, just one more and that is strictly non-linear in nature. It is connected with second degree polynomial and second degree equations. Now I'm not going to explain now in detail in mathematical detail what that means I just want to show you that by solving those quadratic equations you get two square root functions which are the same up to a sign in front of them and the square root function is a typical model of a non-linear reduction lens. Just do a little computation with me. If you take square root of 100 what is it? 10 you reduce quite a bit by factor of 10 but now take square root of 10 what do you get? Three point something which is also reduced but much less. In other words here you see it depends on the position in space how much square root reduces and the fact is that if you take such a second degree polynomial and solve it for x you get two square roots they're the same up to a plus and minus sign in front of them and we now use those square root functions as functions of the plane. That's a mathematical extension which goes under complex numbers. I will not go into that. It's not too hard but it really doesn't matter. There is a way to extend the notion of square root to points in the plane which are given by coordinates and that's what we want to do now but you understand square root reduces and another point which I want to bring over is the equation depends on a parameter C much like the feedback system dependent on some outside control and what we will do in a moment is we will change this parameter again and again. Well, I guess at this point I will try a last computer experiment and hopefully this one works. It doesn't. It does not. So I have to inform you that's life demonstrations with computers. I have to inform you what we will get. Applying the square root reduction length systems we get structures as complicated and fractal as the ones which we obtain with the linear setup. Here's just a selection of eight different structures. Each of them is obtained by one particular setup of a two-length system machine and the change in the picture is obtained by just setting the parameter which is a number into a different position. So you should imagine I have this copying machine I have a dial at the machine I set this dial into some number and then I let the machine go and out I get one of those pictures and then I set two different number and I get a different picture, a different picture and so on. I want to show you one of those in some enlargement because that will become important at the variant. And now I think I've talked enough for a moment I want to show you the variation of these pictures which are known as Julia sets in a little animation from the videotape. Can we have the videotape on please? It ends on parameters that can be varied. The competitive situation and the Julia set change accordingly. A confusing diversity of connected forms that disintegrate into dust and then grow back together again is the result. How can we get a... Cut the tape please. So that far for this animation what you have seen is just some 3,000 different images linked together in this animation and each of them, if you forget the color the essential structure is obtained by just running an individual copying machine designed and built as I explained. So there's an infinite variety of such pictures and here I want to introduce the infinite book the book of pictures. Each page in the book has one unique image which is obtained by running the copying machine with these two square root functions and the different pictures are obtained by just changing this number C in the lens systems. So we have an infinite number of pictures each of them very complicated, a fractal and if we compare the different pictures they all look quite different from each other. And the question which I want to now look at is a mathematical question. A very typical mathematical question is there an order principle in this infinite variety of complexity. In the terms of a book you might ask are there chapters in the book? This picture goes in the first chapter this picture goes in the second chapter and so on. Are there natural chapters in this picture book? The answer is yes and the answer is based on a beautiful and deep result which goes back to Julia and Fatou around 1918 and it says yes indeed there's only two natural chapters. The first chapter collects all images which look like the picture on the left and that means pictures which are one piece mathematically connected. And the pictures in the second chapter look like the picture on the right. A cloud of individual points. Mathematically speaking the cantor set. In other words here you can go in between and in between and in between everywhere. A cloud of individual points. These are the two kinds of pictures which are possible and nothing else. So in other words there will not be a picture produced by this setup which has one piece here, one piece there and another piece there. Three pieces that's impossible. It's either one piece or infinitely many like in a cloud. This is a very profound and beautiful result as I said due to Julia and Fatou. This result was known to Mandelbord when he carried out his historical experiment in 1980. Now you should see this is really a new experiment, a qualitative experiment in trying to see to visualize the order principle. And what he did is this. He picked a plane and each point in the plane can be viewed as one of those numbers C which you dial into the copying machine. So if you pick one point here like this you can dial that into the copying machine, you can run the copying machine and out you get a picture. And this picture will be either this or that. What Mandelbord did is he colored the points so chosen, black or white, depending on whether the copying machine picture would turn out to be one piece or cloud of points. Now I didn't tell you how the computer is able to decide this question. It's not a very decidable question whether you have one piece or not for a computer. There's some other very beautiful mathematical result behind this which gives you a computer algorithm for that decision. Don't mind what that is. Here is the picture which was obtained. The black picture, the black shaped object collects all the parameter values for which the attached copying machine picture is one piece and everything which is outside collects the points for which the copying machine picture would be a cloud. So this black shaped object, now called the Mandelbord set, is the order principle in this universe of pictures in the quadratic dialect. So that was the principle discovery and then mathematicians came in and investigated the properties which they saw in that picture. You see there's all kinds of structure and I shall tell you every single piece of structure which you see here can be interpreted mathematically and the best way to say how it is interpreted would be to say it's like a grammar for a language. Each of the little structures is a paragraph in a very beautiful grammar. So let's look at this grammar a little bit deeper in our next tape segment. Tape on please. Can we have the tape, Pat? Here we have chosen slightly different colors for the Mandelbord set and the colors outside are the result of delicate experiments which measure the distance of a point to the Mandelbord set. In fact, contours of the same color identified points of the same distance to the Mandelbord set. In a similar way, later color renderings will show equi-potential curves of the Mandelbord set. The current magnification factor of this zoom animation can be found in the insert in the upper right. As you have seen in this animation, blowing up this structure, this mathematical construct reveals more and more complicated structure. What does that mean? Is there any way to understand this? There is. And I want to show you what it means that we see more and more structure as we blow up the Mandelbord set. There's a very deep and profound reason for that. And that is one of the current mathematical results in the attempt to understand these structures which I want to introduce by another and final question. Remember what we were doing is we were looking at copying machines which were set into different positions, producing pictures. We would gather them in a book. And the final question which we want to ask is, is there something like a table of contents in that infinite book of pictures? Now, I should admit, this is not a question which mathematicians asked right away. They were led to this question by experiments, by seeing some very strange and remarkable phenomena which turned out to be the answer to that question. So again, what is the question? The question is, is there a table of contents to this infinite picture book? And why is that a remarkable question? Well, if there is a table of content, it must be a picture itself. And that picture, this one picture in the table of content should contain information, pictorial information about all the other pictures which you obtain by the copying machine. And I think here you see how wild and courageous this question is. So what is the answer? Well, you have all seen the table of contents already. Here it is. The Mandelbrot set itself is a table of contents of the infinite picture book. And I think that explains why we must see as we blow up the Mandelbrot set, change over change over change, because only in that way it can store as an image storage all these different pictures. But you would say, if I look at the table of contents here and the typical page in the book there, I do not see any similarity. Maybe one needs one pair of mathematics glasses to see that. That's not the case. Let's do a final slide. In this final slide, you see the Mandelbrot set. You see a crosshair. The crosshair fixes a point very close to the boundary of the Mandelbrot set. And around the point of the crosshair, we blow up. And what we see is a double spiral. And incidentally, the blow up factor here is some 100,000. We were used to that, seeing the animation. Now let's pick this number C, which the crosshair fixes, and let's dial it into a copying machine. Let's run the copying machine and out we get a picture. Here's a picture which we get. It's one picture of infinitely many in the picture book. But you can see this real well, so let me show you an enlargement of it. That is a picture obtained if you dial in this particular number C. You still don't see any similarity, but we are not done yet. Here's another crosshair. Around the point of the crosshair, we blow up. We blow up essentially by the same blow up factor of 100,000, and out we get essentially the same double spiral as in the Mandelbrot set. And in this sense, the Mandelbrot set stores the principal information about this particular page very effectively. And in that sense, the Mandelbrot set is a table of contents of all the pictures in the book. And if you ask me how can one prove such a thing, I have to admit the mathematical proof only is known so far for a particular selection of points on the boundary of the Mandelbrot set. However, so many that one mathematically says a dense set, but there are many other points for which computer experiments reveal this property. However, mathematicians still have trouble to understand that. Let me now go to two more video animations very briefly and then conclude. The first one shows you that principle of the image storage of the Mandelbrot set in an animation. Let's have the tape. But also in our knowledge about the Mandelbrot set. One fascinating property is the similarity between Mandelbrot and Julia sets on a very small scale. To demonstrate this, we show an extremely small part of the Mandelbrot set. This picture looks almost identical, but it represents a part of a particular Julia set. Zooming out the difference between the two sets becomes apparent. The sequence of images is now running backward. That is, both objects appear again under increasing enlargement. Thus, in a certain sense, the essential structure of this Julia set is already contained as a compressed replica in the Mandelbrot set. This is no isolated case. At any point along the boundary of the Mandelbrot set, under extreme enlargement, we can find a structure that looks the same as the corresponding Julia set. For comparison, this can be seen in the window. The Mandelbrot set is, so to speak, a graphic dictionary of all Julia sets. Not only does one point of the Mandelbrot set determine one of the experiment's parameters, but its neighborhood also shows us a picture of the result. By contrast, a further example shows a point with deviating behavior. Here, the similarity is only of a temporary nature. And with this, we have already reached the present limits of the mathematical understanding of the Mandelbrot set. Well, it's time, sir. So much for this animation. Let me now come to the conclusion. The last video animation which I want to show you is an attempt to visualize the electrostatic potential in an animation. The Mandelbrot set can be viewed as a piece of metal. You can charge it electrostatically, and you can measure the potential outside the Mandelbrot set. That gives you some number. And that number will be very small if you are close to the Mandelbrot set, and it will increase as you go away from the Mandelbrot set. And now what you can do is you can color code this number. This is exactly what you see in most of the color images. But you can also take this number, which is small, close, and larger, away from the Mandelbrot set, and view it as a height profile. And to make the visualization a little bit better, we have just turned things upside down. So rather than increase, we decrease. And that is the last animation which I want to show you. I want to thank you for your patience and that you accepted the computer failures. I'm very sorry for that. Here's our last animation with which I want to close. Please. I want to add that the music is obtained by playing the algorithms, which are used to generate the images. And the compositions are by Martin Gochtner, a friend from Stuttgart, Germany. Thank you very much.