 I now have the very great honor of introducing our first speaker, Dr. Benoit Mandelbrot, the founder of Fractal Geometry. Dr. Mandelbrot earned degrees in engineering and science from a coal polytechnic and the California Institute of Technology, and a doctorate in mathematics from the Faculty of Science de Paris. He worked at the National Center for Scientific Research in Paris until 1957. In 1958, he moved to the United States and joined the research staff at IBM. Since 1974, he has been an IBM fellow based at the Thomas J. Watson Research Center, and since 1987, the Abraham Robinson Adjunct Professor of Mathematical Sciences at Yale University. During his career, he has held a wide variety of visiting positions, including those of visiting professor of economics and research fellow in psychology at Harvard University, lecturer in electrical engineering at MIT, and visiting professor in physiology at the Albert Einstein College of Medicine. He is a fellow of the American Academy of Arts and Sciences and of numerous scientific societies and an associate of the National Academy of Sciences. He has won many honors, awards, and medals, including the prestigious Chevalier, the Order of the Legion of Honor in France. A true polymath, he is the author of over 100 articles in professional journals, ranging over a broad spectrum of disciplines, including communication and information theory, the structure of language, thermodynamics, probability, economics, turbulence and fluids, hydrology, geology, cosmology, and art. He has written that, quote, it had seemed too many that each of my investigations was aimed in a different direction. But this apparent disorder was misleading. It hid a strong unity of purpose. Against odds, most of my works turn out to have been the birth pangs of a new scientific discipline, close quote. In 1975, he laid many of the foundation stones of the new science of fractal geometry in his essay in French, Fractals, Form Chance and Dimension. And in 1982, he published the Fractal Geometry of Nature, his marvelous manifesto and casebook of the new science. He is currently working on new books on multi-fractals. Ladies and gentlemen, I am delighted to present the father of fractal geometry, Professor Benoit Mandelbrot, to speak on the fractal geometry of nature and chaos. Thank you very much. I'm delighted to be here and very much honored by being asked to speak first at this meeting. In a certain sense, the topic of my lecture will be followed by that of Heinz Otto Pagan who closed the meeting, which puts me in a very awkward position of trying to tell you about fractals without knowing what Heinz Otto is going to tell you at the end. But let's have been old friends and I'm sure that everything is going to be fine. Now the way I try to organize my lecture is the following. First of all, I would like to introduce you to a large number of fractals. Fractals are geometric shapes and to understand geometric shapes one must see them. It's something very fundamental, very often forgotten and forgotten at very great loss that geometry must have a visual component. And so I was hoping to show you a large number of slides. The best way to do so is going to be to play for you a small part of a program which some friends of mine and myself have organized last April at the Guggenheim Museum. That will come after a brief introduction. So most of the talk will be in terms of projection slides and so on. I'm very worried about the people sitting here and here because they're going to see nothing. So if you can move closer, you'll probably be better off. But that is just a matter of advice. Now, well, so let me begin by the beginning. May I have the first slide? Oh, I see. Can you see me a little bit less? Yes, thank you. Well, I begin by this quote from Galileo, which is something quite marvelous. At the dawn of science, Galileo Galilei described a great book of nature as being written in geometric language. And this language, he says, is made of characters which are circles, cones, et cetera, without which one earns in vain a dark labyrinth. Without knowing these characters, science is impossible. On the basis of these characters, Galileo built the beginnings of mechanics and therefore of science. These characters are borrowed from Greek geometry. And Greek geometry had an interesting episode in its development. If you read, as I did once on some of the advice, The Life of Marcellus by Plotarch, you find a passage in which the following story described two men named Arquitas and Eudoxus had applied the geometry to the science of mechanics and showed that you could, with understanding of shape, understand the laws of nature, and at the same time that you could use the illustration of geometry presented by mechanics as a help in understanding geometry. Plato was indignant at this situation. He said that was a shame, a disgrace, a pure science of geometry should put base usage as to use mechanics, some vile usage, and that a proper geometer is a person who does not need the concrete illustrations of mechanics to do his job. Well, so you see that they were having attention for a very long time mathematics between those who believe in purity, in absence of input from the eye and from science, and those who believe the contrary. Well, clearly Galileo believed the contrary. Now what is the geometry he was referring to? Here is one example of the geometry. It happens to be the IBM Research Center, which I show here not at all because the marvel is building to live in because it's not. It's a typical modern building, but it exemplifies one essential law of geometry in our everyday life. Euclidean geometry is a geometry of shapes around us which are made by man. Man's works are very typically flat, round, in general follow the same very simple shapes of classical school geometry. Now we all know by experience that the world around us, the questions which are primary in the sense of being the first to be asked could not be answered by this geometry. For example, mountains are too complicated for that. And let us look at this mountain. It is not at all a real photograph. The previous one was, this is not. It's completely mathematical forgery, computer forgery, mathematical formula. Mountains are more complicated than any shape in Euclidean geometry. Clouds are more complicated than any shape in Euclidean geometry. Therefore described in nature around us, we have to do something else. We have to go beyond Euclide. How far can we go? Well, if you go too far, you get lost in excess of generality. The two veins of science are lack of generality and excess of generality. There's a certain level of proper generality which is necessary in order to things right. And the geometry which is able to include these shapes that does exist now, it was put together by me primarily in initial stages 15 years ago out of very many pieces which have been around for a very long time. Like everything in science, this has very, very deep and long roots. But they have been a turning of roles. Objects which were viewed as being very, very far from physics turned out to be the proper tools for studying physics. So that is the aspect of my talk which is going to be geometry of nature. And I will describe some more specific examples after this little show. Now about chaos. It's a topic which other speakers are going to describe but I would like to mention immediately that the proper geometry of chaos is the same as the proper geometry of the mountains and the clouds. It is something quite marvelous because might have happened perhaps that different geometries would have been needed to follow that of Euclid, but it's not so. Fractal geometry has two roles. One to describe again the shape of mountains and clouds. Mountains as I wrote once are not cones. Clouds are not spheres. Islands are not circles. Rivers don't flow straight. This same geometry is the proper geometry of everything which is geometric in the state of chaos which you hear mostly from the other speakers. And I would like to show you a few examples of the shapes in which I encountered in this context. This is a very, very magnified version of a set to which my name had me attached. It's magnified in ratio of Avogadro's number. Why that? Well, because it's a nice number. 10 to 23, because as it turned out, this number was a good opportunity for testing the quadruple precision arithmetic on IBM computers. It's very amusing to be able to justify fun and science on the basis of such specific roles. If the whole Mandelbrot set had been drawn the same scale, the end of it will be somewhere on serious. It's enormous magnification. Now you will see later something about that and Hans Otto I'm sure is going to talk about a great deal. Now this shape here is a variant of the same set corresponding to slightly different formula. I will not repeat the formula. I would like to show this shape simply to comment on something which is one totally amazing and extraordinary satisfying aspect of fractal geometry. It is that shapes which are initially developed for the purpose of science, for the purpose of understanding how the world is put together both statically and dynamically, statically in terms of mountains, dynamically in terms of chaos, strange factors, et cetera. The same shapes are perceived by many people as being beautiful. This shape here was not intended to be beautiful. It was just an exercise for going through to try to find something else which we could study with reasonable facility to justify investing time in it. And it turned out to have many features which surprise us to this day. For example, it seems to be three dimensional. In fact it's not, it's completely flat. But to my eye there is like a piece of leather which comes up and goes down, cast a black shadow to the left and the edge of it also casts a shadow. I'll mention that simply as a kind of question. Why are these shapes beautiful? What did they tell us about our system of perception? Because experience has shown us that the same features are seen in them by many different people in some cases. In other cases different people read different things in these shapes. The shapes don't mean what people read into them. Therefore there is a very great deal of understanding of this matter which I think we are going to come in briefly towards the end. Well, you also see shapes of this form like this which is called the snowflake curve or the Koch Island has many names. Was discovered by a man named Helge von Koch a professor of mathematics in Stockholm in 1905. And these shapes play a very important role in the faculty geometry because they're easy to study. They're easy to study but in a certain sense they're very finite. You must begin with them but that is not where the fun begins, the fun begins beyond them. And so how to present the fun? I would like to play a small excerpt from this video. Let me explain how it started. Charles Worrien is a very famous composer in New York very highly very well known police surprise winner, et cetera, et cetera. He called me many years ago and suggested we meet and we had a very good time talking about music, about fractals, has a very, very good understanding of science which very much surprised me and impressed me. Then last summer, last winter, we discussed the possibility of doing something together. What came out was that Charles Peace New York Notes which we see here described was played at the auditorium of Guggenheim Museum in New York on the right side of the stage. On the left side of the stage, we, that is Richard Voss and others whose names will appear in the credits, are prepared to gather a collection of slides. Now so I'm going to start about the middle of this piece and it is not the whole piece but you will and please take it in two roles. One which is just a collection of slides shown very, very quickly after each other and merging into each other better than I can do with this carousel and at the same time as a preliminary to show discussion of fractals in music which I will have after it comes through. So may I have now the video? May I have this slide down please because it will help the video with this slide. Could you please cut the tape now? I'm sorry to interrupt in the middle but we must go on. I will explain to you later about some things about it but now let me begin by repeating again the music is by Charles Warren, the musicians who are playing are, well, many of you must know some of these names. Fred Cherry is the very famous cellist and director of New York Lincoln Center Chamber Music Players and this piece is going to be played there again next March on 8th, 9th and 10th of March. It has become something of fixture in New York musical scene but I have no time to continue for it. I will explain a few minutes how this union of music and fractals came up but let me now come to business. You have seen many, many shapes, all of them computer generated. Each of them had the following characteristics. It had a very simple formula, formula reduced to its essential states, one line, two lines perhaps but out of this formula comes this extraordinary wealth of structure that you have observed. This wealth of structure was taken up by a very fine computer people whose names I'm going to mention momentarily when I showed the slides again and who rendered it. The effects of light and so on are state of the art in computer graphics but are not part of the main point. The main point again is the coexistence of very great simplicity of formula and very great complexity of shape. And so here we are in the middle of this problem, this kind of quandary which science had faced for so long that the geometry was extraordinary effective, extraordinary powerful as a tool in the sciences by geometry, I mean Euclid. Yet everything around the experience was different from Euclid. If you want to go beyond Euclid and create new shapes which are able to comprehend the complication of nature, we must in a way go by small steps. And one of the central tools of science is the idea of invariance. The idea that somehow things are complicated but not infinitely complicated, they're complicated but orderly. In the case of fractal geometry, the great invariance is self similarity, self affinity, self alightness, self symmetry in this sense. One has a complicated reality which has a great deal of structure, the structure being the fact that each part is in some sense the same thing as the whole but smaller. Now anybody who has been attuned to art for over the centuries realizes that this idea had been very common in the arts. The idea that well in Indian paintings very often there are small Buddhas on the robe, a big Buddha, and yet smaller Buddhas on the robes of the smaller Buddhas and so on as far as a painter's brush could do it. In, again, in Buddhist writings, there are stories of big palaces which are such that each room of this palace is like the whole world and in it there is a small part which is like the palace itself and that small part is the room which is like the room in which the speaker or the man who tells the story sits. And coming back to Western civilization, certainly in Aristotle, there are many such echoes, in Leibniz they are very, very strong. Leibniz had the idea that in each drop of dew, a whole world resided. If you look carefully in the spring of the leaf on a drop of dew, you will see a sun with its planets, with their beings, their leaves, and their drops of dew, et cetera, et cetera, at infinity. Therefore, this idea of repetition, the idea of the palace like the whole is one which is very much fabric or are thinking as humans for a very long time. The only trouble with it and a very major trouble is that was scientifically a poor idea that is what applied in context where it doesn't hold. It is not the case that an atom is like a sun with planets, only in a very, very crude way, otherwise very, very different. Therefore, much of the thrust of science over many years has consisted in fighting this, how to say, natural sliding into the notion of self-similarity, self-alikeness, which humans are very prone to entertain. Well, a fractal geometry is based upon the notion that there are some fields of endeavor in which the sliding down is not necessary. And so the basic invariance of fractals is the one I described. I'm going to give you some examples of it momentarily. Now, which is the simplest example of it? This is one. It is due to a man named Shapinski. I called the Shapinski gasket as something of a joke and the joke has stuck. It's very obvious by looking at it, it's made of four parts, or three parts, blue, yellow, and red, each of which is exactly like the whole, except it's twice smaller. Well, Shapinski defined this shape for some purpose, which has been forgotten, rightly because it was not very important. But the main fact about it is it is so simple that in a certain sense it is almost like Euclid. If you come to think about it for a few days, you know about everything there is to be said about it. It's a shape which is rather poor. The formula which consists in taking the middle part of this triangle and so on and so on at the infinitum is elementary and there is not much you really find by studying deeply. So the impression may prevail that if you take some similarity, it's a barren and not a fruitful idea. But to show that it's not so, this same thing in three dimensions, to show it so, let us go a little bit further. First of all, we go to some plants. Some plants in fact have a surprisingly large degree of self similarity. This branch here is completely artificial, computer generated, but each branch is exactly the whole except it's smaller. Well, very many plants are very nearly self similar. And they should be because they consist in branching, branching, branching. Each stage of branching is ruled by the same rules therefore the same things happen. But the bigger the plant, the bigger the thing that happen. That's kind of very rough idea what makes self similarity arise here. It must be so. One often fears that self similarity in this fashion would hold only for a few levels that if you go in greater detail, the branches stop branching, they end up, they haven't ended, it's true. Like very often in science, you have a principle which is simplifying principle of understanding, it does not apply if you go too far in the very small or too far in the very large. Sometimes it does, sometimes it does not. Very often it does not. Now let's look at this shape here, which is a real one. Second photograph of a real object, you may recognize a species, a variety of cauliflower called Romanesco, which is characterized by an extraordinary level of self similarity. If you look at each of the knobs of it, they look absolutely like the whole. And then they carry themselves knobs and so on. Well, the first reaction of scientists looking on such shapes was to focus on all the spirals which exist in it. And there's quite a lot of knowledge and botany about the way plants spiral. Fibonacci sequences play a role in that which is very interesting. But what is more important for us here is the fact that each part of it over five levels of separation which you can see and then many more levels you cannot see but only except magnifying glass or microscope, you find the same structure repeat again and again. Therefore, this notion that somehow self similarity would be a dead thing is not true even in this case. But the most important things go further. The most important thing go when we add an element of unpredictability or an element of nonlinearity. Actually, nonlinearity, which is chaos and unpredictability, which is chaos and different in all the sense of the word are very intimate linked. I will say a few words about that if I think of it. But that is what I'm going to come to in a minute. But before that, I would like to say a few words about music. The reason why Charles Warren came to see me is the following. He said that for many, many years he had been worrying in his own mind about difference that exists between noise and music. Clearly, he would say, listening to something, this is music. It's very bad music. Very bad, but it's music. And listening to this thing, he would say, that is noise, a pleasant noise, but it's noise. Now, what makes a difference in my mind, that he told me between these two? And he was quite incapable of formulating his ideas on the subject. Then he looked, he saw my book and saw the key to it. Now, the one part of the idea of fractal geometry as language, which I have not yet emphasized enough, fractal geometry is a language, a geometric language which can be used to understand, to organize our experience and to go further, to explain our experiences. For Charles, this was a matter of organizing. He always felt that a long piece of music must be in a way composed. That is, if you have 30 minutes of music, you have a fast, a slow, and a fast movement. Each movement will have a loud, a soft, and a loud part. Each part will have a clarinet solo, a violin solo, some other solo, and so on. There must be changed variety at all scales. And that characteristic, he told me, was to him the principal criterion that distinguished music as an organized activity from noise, which is organized differently or is not organized at all. And therefore, he asked me whether he had any kind of inkling of that. Sure enough, had very strong inkling because my friend Richard Voss, who is the author of many of these pictures, I'm going to identify for you very shortly which pictures he did, who is actually a Minnesota boy from St. Paul, not far from here, but who left this state for other places on the east or west coast. Dick Voss had, in his PhD thesis in Berkeley, been playing with one of Raph Noyes. What is one of Raph Noyes? Well, it's a noise fluctuation, which is between hum and the brown motion. I'm going to come to it momentarily in the next slide. It's a form of fluctuation which is very, very common in physics and very little understood in many ways a very great mystery. So Richard Voss was trying to understand one of Raph Noyes and he was analyzing all kinds of things around him to see whether perhaps they were or were not one of Raph Noyes. He found that music, the notes, the loudness, all kinds of characteristics of music are scales longer than a sound that is not a matter of analyzing the sound of a cello or sound clarinet, but the musical part of it, all these characteristics, if properly analyzed, gave one of Raph Noyes's. What does it mean? It means precisely what Warren was telling from Musician's viewpoint. That is, organized music in western culture had this very strong structure was through Bach or Beethoven of the Beatles because the Beatles are just like academic in terms of the writing as Bach or Beethoven. They follow the same rules. He went to analyze some records from a store selling ethnic music from Africa. The same thing was true, one of Raph. He went on to analyze Boulets and Stockhausen and they are not one of Raph. Now, a very general feeling that Boulets and Stockhausen were different from Bach, Beethoven, and Beatles was confirmed in sort of objective fashion. Now, whether you think that it is stronger than whatever you believe before is not a matter. Sometime later, I met another musician who was a legatee, George Legatee, and he told me exactly the same story as Warren. In fact, he went on saying that all musicians have this feeling that some things are music, others are not, and in Legatee's opinion, the feeling is based upon whether there is an element of self-invariance, self-symmetry, self-affinity, more specific in that structure. When there is, it's perceived as being music, good or bad, when it is not, it's perceived as not being music at all. Well, in the music which you heard in that short video, a part was computer generated. Given the scale of Warren, it was not obtrusive. It was not something which stopped the action and said, here I am, I'm a computer generated piece. But it was eight musicians, the seven whose names were shown on the credit line, and an eighth which was a computer generated part. It was amazing speaking with the audience after the performance to realize that most people did not feel that a new instrument has come in. And that new instrument was noise generated according to very simple rules in which it was made sure that self-similarity was maintained. Even at this very crude level, Warren felt that this added an element to his music. Well, let me continue faster with what the fractals are and introduce notion of fractal dimension, which is very fundamental in this context. Now, first of all, self-similarity. You see on top an element, a segment is divided into pieces. If you divide into five pieces, each piece is one-fifth of the whole thing. Then you see a square, a cube is divided into pieces, and the square is divided into four pieces. Each of them is smaller in ratio of two. Therefore, this ratio, which is written both letters in the bottom, d equal logarithm of n divided logarithm over n over r, which is log of number of pieces divided by log of the reduction ratio. This ratio is something which is identical in these cases to ordinary dimension. It is one on top, it's two in the plane three in space. Well, starting from this very broad intuitive idea, which is called similar dimension, mathematicians have developed a large number of notions dimension. One of them is called the Hausdorff-Bezikowicz, because people created it, others, another called Minkowski-Bouligon, et cetera, et cetera. There are ways of implementing this idea of dimension in such a fashion that instead of representing what we ordinarily know as dimension, which is a point dimension zero and the line dimension one and the plane dimension two, et cetera, you can interpolate and have shapes which are in between a line and the plane and have dimensions which are, for example, 1.4, 1.5, 1.7. Well, I would not like to emphasize this dimension, I give dimension because for many people, it's mysterious and perhaps overly mysterious. The main thing is that this is one of the basic technical tools of the study of these shapes. Now, let me proceed in the case of fractals instead of obtaining dimensions which are again integers obtained fractions. So on top, you replace a straight interval by this zigzag curve and you get the snowflake curve in the bottom. You take this square-ish zigzag curve, you obtain this other shape. You take this other shape, obtain a space-filling curve. In general, you can manipulate shapes by these very simple recursive mechanisms which give you exactly linear reductions and somewhat dull structures. But I say that the two variants of it, I have no time for, set affinity and multifractals, very important but beyond the time I have at my disposal. I said before, if you want to get to really interesting structures, you must go beyond exact linear set similarity to either randomness or to nonlinearity. And from now on, I'm going to study first the randomness and then the nonlinearity. First of all, randomness. On this collection of pictures, each of them going from top to the right is the reduction of preceding one. That is a blow-up. That is, you take a small piece which is marked by white square rectangle. I hope it's visible from where you sit and you blow it up to see greater detail. You add more detail, more detail, more detail. Now, the way you sit, you sit because of the writing is small. I think the impression you must be getting is that there are nine pictures here from all is the same, about the same but very different in detail, but they are of the same kind. And that's the main point. The point is here that by having an element of randomness of unpredictability, total unpredictability added to this construction of recursive adding of detail, one can obtain an amount of richness of structure which was totally beyond any expectation. The same way as again in the case of music by varying loudness or things, you get the amount of detail which was well beyond expectation because nobody could believe that so much richness of sound could be obtained by simple rules. And in this case, of course, the mechanism was chosen in order to represent something about the shape of mountains. And so let us now give a pollute to the second thing, the transformation which I have studied most in the case of nonlinearity is one which goes from Z to Z square plus C. Why this one? Because the simplest you could write. More precisely, two which are simplest. If you don't want to have something linear, you either take a square or take the inverse and I've studied both. The one with a square turns out to be richer and more fun to work with than the other which is why it became so widely popular. Now the first person who thought of this Z square plus C transform, well, it must go back into very dim past. In the 1870s, it was something which was not a new question in a certain sense. But certainly in 1904, a man named Fatou did a great deal about it. Then for many years, it was enormous interest in the study of it only on real access when you get from X to X square plus C. And the work of Michel Fangenbaum is at the center of this investigation of the transform X going to X square plus C. Well, what I did was to work on that in a complex plane Z and I'll come back to it a bit later. Oh my goodness, the music is completely out of line. So I'm sorry, I skip it because, well, there must be a conference, especially chaos, one element of randomness in slides. So you just saw it. Now we go to this mutation of mountains briefly to give credits and then I'll proceed to something in physics which is very musical and expected. The thing which, the question which in a certain sense is a center of what fractals are is the question of how long is the coast of, well, I say Brittany or Britain because it's very nice both in French and in English. The coast of anything. This question has been at the very core of the origins of geometry. That is if one reads about the origin of geometry, that very much of course due to the Greeks, an extraordinary phenomenon that geometry was born in Greece and nowhere else. And very much influenced by navigators who had gone west to explore Sicily and Sardinia and were very much at odds about how large the two islands were. Because in one hand it took less time to go around Sicily than around Sardinia. But it took less time to cross Sardinia on foot than across Sicily. So they're arguing very much, political parties, the parties saying Sardinia is bigger and the parties saying Sicily is bigger. And eventually the intellectuals came in and distinguished two things, the area and the perimeter. So they said what happens in Sardinia is an island which is smaller but has much more complicated coastline. And how they measure the coastline, well like any sensible Greek would do. They would just take a small boat and navigate, circumnavigate the island and see how long it takes to go around it. And they found indeed that in the case of Sardinia the coastline is very long, case of Sicily it was not very long. But if you come down to it, the whole thing was ridiculous. Because why a small boat? Why not a big boat? Imagine this area here is completely imaginary but it's very much like a real coastline. If you circumnavigate it with a big boat avoiding the little islands because you may get a ground then you cover a small distance. A small, very flat boat would follow it very closely and get you a longer distance. If you walk around the coastline with seven big boats you're going to get a short distance. If you walk on coastline with ordinary boats you'll get a longer distance. A mass will go into details where you have just two steps and an even longer duration. So if you come down to it, the idea of length, which seemed to be totally obvious, solid, which every teacher of geometry had no doubts about, was in fact very, very soft. And then you go and see in practice did people make mistakes about length? They made terrible mistakes. For example, if you look at the Asicropedia of Spain and Portugal, the length of the common border in the countries is described by Portugal as being 30% longer that what Spaniards say. That's the same line. Drone on it and people have been killed and I don't know how many wars to determine that the border is here and not here. And yet the Portuguese say that 30% longer border. Why so? Very simple. Portugal is a very small, much smaller country than Spain. Classrooms, the maps where people measure the coastline are not the real terrain but maps. Maps of the kind you have in schoolrooms. In the schoolrooms, Portugal gives maps for much more detail than Spain because smaller country. Therefore, details which are not visible on Spanish maps are visible on Portuguese maps. So you see there is an element which is very important of what the structure, the language you describe something with does to your description. By describing frontiers by the language of Euclid or Negeometry, you hid this very fundamental fact. Now let me run very rapidly. This is an relief obtained by very, very trivial things piling pyramids upon pyramids. The most childish model imaginable. A mountain is not a cone but if you pile enough cones or pyramids on each other you get these very lifelike things. Now these mountains which you saw before they are due to Richard Voss again from St. Paul, Minnesota. This one is a composite of different things again due to the Voss you saw it in the video. This is something which is an imitation, if I can call it, which was produced in a science fiction film. There's some people who say that there's nothing against the fractal geometry except the fact that it leads to imitation science fiction films. Well that is meant to be put down. I think that actually it's very amazing that something which was so sophisticated and so esoteric and so difficult before I had a notion that might be a good way of describing mountains turned a few years from being that into being popular exercise but something happened on the way. That is to make these things faster and cheaper they reduce the element of self similarity of self aliveness that this shape has as compared to preceding one. And you see all kinds of defects you see kind of V in the bottom of the enough light. You see a texture of crumpled paper, very, very visible. Here is a case in which something which was viewed initially as being purely an abstract mathematical idea, the idea of invariance being essential that is a shape being run by its own invariances. If you weaken the invariances you make them less powerful then the shape becomes wrong. I had never expected myself that this would be so visible. And again for those of you who know these references there was a man named Felix Klein who in 1870 had a program of science which was based upon the invariance of both mathematics and physics. These pictures here which I've seen are due to Ken Musgrave who's a student of mine at Yale. It's part of his thesis. And this one of course competition between Ken and Dick. He wants to show that he can do Earth just as well plus the clouds which Dick didn't have. This one is a new development which is very interesting. James Bardeen who is a physicist at University of Washington, a cosmologist and relativist of a great renown is also a computer nut. And he has his own Amiga computer and he read a paper of mine in which I expressed a very great dismay at the fact that all these previous mountains however impressive they were from when seen from an angle rather low on the horizon had something very wrong with them. We could have either mountains without rivers or rivers without mountains but to put it all together I had to do something very artificial. And so Bardeen came to think and reasoning very much like a physicist would putting minimal amounts of further information about how the relief is formed. He got some pictures. I'm going to show to them with some apologies because they had done a very small microcomputer. It's remarkable how good they are for it but they're not the quality and finish of the others. So you obtain these effects that are not put in by hand. This relief was obtained by an algorithm which tells you a possible scenario for the formation of relief. Geologists I know are very impressed by this scenario because they think it may well be the scenario that created the Earth relief. But if you want to test these scenarios you must see what they actually give. And here the eye is our tool. Some scenarios if tested by being implemented in pictures vanish instantly. They make very promising a priori but once you see them they are just wrong. This scenario wins on the basis of being likely end of being very reasonable. These have seen our clouds by Richard Voss. Some flowers by Prushin Kiewicz have no time to explain them very much. And I'll go to a chapter of Equiculture and Physics. Well, this cartoon appeared in American scientists. We did the whole room over in fractals and it is very misleading. And I put it here both because I was so amusing because I want to emphasize it. In this cartoon the straight lines like a table and everything of how to say human manufacture has remained straight in Euclidean. It's only the decoration which is fractal. And the two are sort of separated. There's one and the other, they're not mixed. But in every investigation I've been engaged in they are mixed in completely in extricable fashion. Well, let me give you examples just to remind me the matter of distribution of matter in the universe. This distribution is a fractal scaling variant over range which is a topic of great dispute. Some people say it's only five megaparsecs. Other people it's a thousand megaparsecs. The great voids one discovers in between galaxies are a telltale sign of fractals. They are obtained by fractals almost always and to obtain without fractals requires an extraordinary amount of special assumptions. So quite possibly this model to represent distribution of galaxies of very long range. And I mention it here because it's as an introduction to the kind of central issue as I see it of the role of fractals in physics. More precisely statistical physics but in physics in general. Physics has been ruled by some rules which are called the partial differential equation of physics. The Laplace equation, the Fourier equation, Navier-Stokes equation, et cetera, et cetera. As the names indicate they are roughly 200 years old. They arose roughly 100 years after Newton. They are in many ways a consequence of Newton, a development of Newton in 100 years. They have been very powerful, extraordinary powerful, very surprisingly powerful in describing the structure of the world. They suppose a world in which everything proceeds in a very smooth fashion because they demand not only one derivative but two. And fractals typically have none. I have forgotten to make this point the absence of the tangent, absence of length goes with absence of tangent and non-differentiability. So there's a conflict, how does the conflict resolve? How is it that the Laplace equation or perhaps the Newton equation or perhaps Einstein's equation of gravitation give rise to this distribution? In this case there is not much known but in other cases much more is known. This is just a passage to mention that there is something quite extraordinary about surface of the earth that the fractures in surface of the earth have the same rules over extraordinary broad range of scales that the same rules follow for 100 kilometers over Nevada desert and over centimeters. But again it's challenge for geologists, nothing is known. Now I will skip on that turbine, we have no time. I will skip on this because I have no time but just to mention one thing which is that in the case of diffusion of heat which is a very important issue, if you have particles of two colors going to each other, let me try the next thing. So here in honor of the Swedish flag it was done in yellow and in blue. You have yellow and blue particles diffusing into each other. What the theory which I was taught which everybody was taught told us was the thickness of a diffusion layer at one end of all blue, at the other end of yellow and between it was in between blue and yellow and telling us what percentage blue and yellow. The question which one could ask oneself is what the shape of the bond between blue and yellow. This question arose from very practical investigations on soldering as it turns out has given rise to something quite extraordinary, the study of a form which exists between blue and yellow which shows that even in this particular problem the two aspects of smoothness and of fractality coexist. But now I come to the most interesting one because most challenging because of course I'm working on it. It's a study, the structure of dust, the structure aggregation. This here is a photograph of a gold colloid. There are shapes which are little blobs of gold which stick together and which end up by creating shapes which are very complicated like that. The structure of these shapes is fundamental to understand the physics of colloidism, what they do and how they act. One must understand them. That structure is entirely fractal and the fractal characteristics of it, dimensions of different aspects of it, determine the physics. I went very rapidly through another slideshow and percolation, another field in which the whole of physics is determined by the geometry. The reduction of physics to geometry was a great dream of many physicists and of course Einstein has done more on anybody else to realize it in his context of general relativity. But in this very different and very much more modest context the same thing is happening. All the physical properties are reduced, the geometric properties which in turn are reduced to the mechanism that create them. Now how do you explain this structure before? Which there's a model called diffusion limit aggregation due to Sander and Whitten, which I will not explain because I feel pressed by time and this model as in three dimensions gives rise to these kind of shapes. Well, the reason I'm going to now stop on this thing because it exemplifies a question of scientific method which I think is important and which is close to my heart. The very brilliant physicists have discovered this mechanism and have found a way of having particles come and get stuck together and color them by age. So the ones which came last are colored outside and the oldest are colored inside. And after this was done, after the fractal character of this shape was identified because one has seen those pictures and was then able to make some measurements, physicists came back to the natural behavior of physicists, tried to make a theory of it, tried to make measurements. Now they didn't know what to measure really but that doesn't stop a determined person. Many measures were made, it was very difficult to interpret. Many were extremely important, others were dubious and others one didn't know why bother measuring these things. The effort to make a theory has been very, very difficult and very unpromising. Now why is it important to be bothered by theory? Well, because in this case it turns out that in mathematical terms, this shape is created by a Laplacian field. You have an equation which is again Laplace equation, solve it, you obtain what is called the harmonic measure of Laplace equation. Many of you know it, others don't, don't worry if you don't know it. I'm just explaining you a few lines what it does. And in response to the harmonic measure, the boundary of the shape gets modified. So you see something completely new entered into mathematical physics. Before one solved the Laplace equation with fixed boundaries, all with boundaries moving on their own separate will, pushed by a piston for example. Here one solves the Laplace equation in the environment where the boundaries are modified by the solution of the equation plus an element of chance. So the equation itself changes boundaries. And that is the key to solution of the paradox of the coexistence of equations and of fractals. Equations are the smooth part, the Laplace equation is valid outside of this very complicated branching structure. The branching structure itself is a boundary at this fractal. That is the fractals are not contradicted by partial differential equations, don't contradict them, they are created by partial differential equations. Now how does it happen? Well, to do so, we, a postdoc of mine, named Carl Evert and myself, decided to look at these things closer. And so here is a figuration of the different way of doing these clusters. They don't grow from one point out, they grow from a bottom up. That's the way they were shown before. That's the way they were shown before. We solved the equation and showed it. And rendered it. Rendered it not for the purpose of decoration, but for the purpose of scientific research. To be able to understand what's happening, want to see exactly what the potential is doing. We changed rendering constantly, same people, some assistants who work on them, students assistants who work on the mountains and such things, we're working on this physics process. They changed the rendering a little bit. We saw something, we changed again rendering to see whether what we saw was real or not real. And this extraordinary, complicated and rewarding interaction of several people with a machine, without machine, nothing will happen, what people, nothing will happen. And different people interacting, we sort of hunted this potential in its layer. And so what we got was pictures of this sort. Now, this picture here, for reasons which I have no time to explain, and young specialists wouldn't appreciate, tells a great deal about potential. The fact where the potential is, the white regions are the low potentials. And it was believed the low potentials are mostly in the bottom, but in fact they're very much up there. So here is a case where to study this shape of exquisite complication, we really had to understand what it is. All the theories which I told us was in the bottom were just based upon intuition, upon what's obvious, what's natural, but not about fact. In this world, we must learn, constantly modify our intuition of what is important or not. Well, then we had this rendering for different purpose. This was not a rendering of the potential when it was ever rendered, it never went in the deep fjords. It was just to illustrate some ideas about potential. But we decided that potential was something deserved to be studied much more carefully. We turned the picture around, we didn't want to look at the world from the then dry viewpoint, but at then dry from potential viewpoint. And for potential, this thing goes here. Well, let me go briefly because I would like simply to make, not to teach you a topic as impossible, but to give you an outline what's happening. One of my colleagues at Yale is Peter Jones, who is a great specialist of potential theory and had been teaching for several years a course of potential theory on fractals which had emerged out of pure mathematics for its own way. And he had theorems about the behavior potential. He had never realized what the theorems really meant because he had never seen them in action in a difficult shape. We had not heard of Jones' theory because it was very esoteric. But being colleagues, we met, he saw our pictures, we started talking it over. And there is extraordinary meshing between the very abstract mathematics that Jones and Carlson and others are doing on this thing and the physics that we are doing. We are doing the same thing. The same thing can be called very pure mathematics if you go into mathematical details in the right. It is all done called physics if you search for reasons of the structure. Well, let me now go into different mode. The second part of my talk would have been chaos and fractals, but since Heinz Otto is going to present that, I will just go and show you some pictures very, very quickly in very few minutes. So this is a matter of Julia Setsch, picture some comments. Now Julia, it's not Julia, it's Julia. It was my teacher mathematics when I was a student. And when I was 20, I was advised by many people, my uncle in particular, a very influential man in my life, a very great mathematician, to write a thesis on Julia Setsch. I read that book of Julia and couldn't make heads or tails of it. So I told my uncle, I will never, never want to do that and did something entirely different. But learned about Julia theory. Now Julia was a matter with a great deal of a geometric intuition, but absolutely no computers. And in fact, I don't know whether he would have been used computers had he had them because of his personality and the day and his time. But we did, we started looking at computer picture of Julia Setsch, discovered many facts about them which were absolutely obvious and very difficult to prove. And here I would like to make a point about mathematics which is very important. It's very important and also I must say it's controversial because some mathematicians are very much of opposite direction, opposite opinion. I'm those who think that in mathematics there are facts. The facts can be unproven and well described. They're called conjectures, they can be proven and they're called theorems. But the so-called four color problem was a fact of mathematics for very many, long time before it was proven to be correct as a theorem. Many results about soap bubbles were facts about minimal surfaces for 150 years before they were proven. By observation, one can very often get to facts. One can never prove anything by observing. So to achieve mathematical facts, one must take drawing as a tool. The same way as primitive man I'm sure was drawing things on the beach or counting pebbles and obtaining facts about numbers and facts about elementary geometry before Euclid came to put the facts together. Facts have their own life and they can be objected to intuition. And by playing at length with the Julia set and sets related to it, we discovered many facts of mathematics of which some were proven instantly by the competent people. Others are yet unproven and others are proven more or less completely and have generated great attention. So these shapes here which you saw in the video are by Alan Norton who was post-doc with me for a while. These are again Alan Norton's shapes just for fun. These are cut through them and this man on top and this is the set, this picture here as I said was by Voss. This Mandelbrot set magnified 10 to 23 times. In some corners of it it's self-similar. Here is an example of a corner that's self-similar. In other corners it is very far from self-similar. It's a shape which is most conveniently called fractal but it's at the very, very end of fractals. It's sort of at the very boundary of fractals. It's almost not a fractal but certainly very much part of the same theory. That's my favorite signature version of it. You saw these things obtained by modifying the definition of Mandelbrot set by changing some signs and I showed them just for the beauty. And this again would have been a work of art I suppose of embroidery of some sort but it's again mathematical formula rendered rather simply by color code. This one has elements of India which is due to certain very peculiar mathematical properties which we put in the model not to any effort to simulate Indian art. And this which looks like a piece of crystal was the effect of a computer bug. This one was a different computer bug. You recognize in the shape outline of it the outline Mandelbrot set as if you should because that's what it is. The rest was the result of just very bad programming which we preserved carefully and this one again is what you saw before except it's turned around. And so let me sum up again it's best to think of actual geometry as being a geometric language. Language is being judged always by not by itself but by the quality of the works to which it gives rise and ordinary language is judged by quality of poetry and the quality of the prose. Poetry would be a language used just for the pleasure without any usefulness. In this case you have already in short lifetime two forms of poetry. One form of poetry which was just purely visual that so many people would think that is art and would be so involved in understanding why it is so why it looks like art rather than just mathematical diagrams and it has great influence on many painters. In terms of prose, prose itself ranges from very utilitarian. Teflon directories are examples of prose. Railroad directories examples of that also. In that sense, fractal geometry has been extremely useful on a very lowly everyday basis to engineers. Very often you deal with messes. Messes so complicated that even most skilled people say throw their hands up and say I don't know how to describe it, I don't know what to do with it, I don't know what to measure. I'm just lost, it's just a mess, it's just chaos. Chaos in the ordinary everyday sense of the word. Very often by being aware of the new vocabular fractals one knows what to measure and one can separate different chaoses, one can organize them. One can do what is said sometimes just natural history. Natural history is a very noble enterprise. It's not the last word in science but it's very important when you deal with very complicated messes. Now, what is the other kind of poetry? The other kind of poetry is mathematical poetry. Again, the fact that by using the eye, by using the computer to help the eye, by, we find that there is enormous number of new facts of mathematics which we can discover. Those facts can be absolutely useless at the time. They can be very useful to physics like the facts of which we discovered about potential theory are. They vary but mathematics has its own standing. It doesn't have to be judged by usefulness because it is one of the great achievements of human mind. And last but not least I think is mathematical physics which I view as being kind of the great prose of science which is again the great equations of science that you have listed Laplace, Navier-Stokes, Fourier, the diffusion of heat, propagation of waves, the propagation of potential of stars on each other and so on. These equations have sat for 200 years and have had enormous influence on our life. It is remarkable that by focusing on the fractal aspects of the world somebody or another one of my friends has been able to add to these equations new problems of extraordinary unexpectedness, novelty, difficulty and interest and to say one word beauty. So we are in this situation of why it is that these things are beautiful. Either beautiful because of the pictures are so much attractive to us, beautiful or ugly but reacting in a fashion which is emotional to mathematical shapes whereas the ordinary action of most people to mathematical shapes is that of boredom or of disgust or repellence. People like myself who lived in love of geometry, I realize when I was a student who were in a very, very tiny minority. So there is a very long discussion about perhaps we have been developed as species living in a world which full of fractals, the mountains, the trees, the clouds and everything of shapes of this form. Therefore if you see these shapes in a different guise, in conditions where the self-similarity is less obvious than for example for clouds or for mountains but is there anyhow, we recognize something familiar and familiarity breeds how to say relaxation and well that's one trivial explanation. Another much more ambitious would be that our whole perceptive system has been influenced by this environment and that it had been organized to analyze these kind of shapes. Therefore it loves to analyze new shapes of the same kind and there is a great deal which can be said about perceptive system on this basis which of course I have no time to discuss. As to mathematical beauty, it is something so elusive that I would hate to comment about it tonight. But what I would like to end on is just the message of about the unity of knowing and also feeling that the necessities of everyday science, everyday work oblige us scientists to be different when whether we do mathematics because we have certain criteria or do physics different criteria, do chemistry different criteria but in a certain sense the division between these categories are not got given that man made that divisions of convenience. In this context which concerns a very simple, very crude therefore a very fundamental invariance namely self similarity, we have elements of everything of mathematics, of physics, of chemistry and we have again this new element of a static, which is certainly a very great surprise and a very rewarding one. So I just could only say but end by saying that I have found that to be a very strange adventure to be drawn into that many years ago and to have gone from one of these fields to another as the chairman said without ever feeling that I was changing my activities. And I hope that you have liked some of these pictures with me. Thank you very much.