Subtitles of Chaos [1] 1- Panta Rhei Panta Rhei. Everything flows. Everything is movement. This is the message of Heraclitus, 2500 years ago. The air around me contains a large number of molecules that constantly collide like billiard balls. Everything is movement. Even my rocking chair is carried by the Earth in its foolish rush! For years, humans have observed these movements and gradually they have tried to predict them. First astrologers, then scientists, and especially mathematicians. By the way, many mathematicians have cast horoscopes! Predicting the movement of the stars and deducing our future from it is an old dream. In the mathematical world around us, everything is moving as well. Watch this rolling ball. Stop! Can you guess where it will be in two seconds? It continues straight on of course, like this! This prediction was easy! Here it is a bit harder. Stop! The ball is clearly going to hit the edge of the billiard table. How will it continue? The mathematician can calculate the future position of the ball as a function of time. This is already a success for Science. We can even calculate what happens when two balls collide. Gradually, the idea of determinism has imposed itself on Science. If I know the situation now, I should "in principle" be able to determine the situation a bit later. Like here for example. What will be the trajectory of the cue ball? In the game of billiards, the player must hit the two balls with his own ball. Not bad! And here, what will happen? Not bad either! If I read the paper today, am I capable "in principle" to foresee what will happen in the world a month from now? Too complicated! Do not think that this film will help you predict the future! Let’s go back to our billiard table, but now with fifty balls on it. Imagine a perfect table on which the balls roll without any friction. When the cue strikes the cue ball, the movement is ... complicated. Can we predict this movement? Of course we can: the computer that calculated these images did it. But it made many calculations. It's clear that the balls follow rectilinear trajectories and they bounce off each other. There are many bounces, but if we take enough time, or we calculate very quickly, we can predict the trajectory step by step, collision after collision. The computer could easily calculate the position of the cue ball in one hour, for example, but this takes so many calculations that it far exceeds the capabilities of a human being. Prediction in principle, perhaps ... but in practice? Let’s move one of the balls a few centimeters. Look. . The two pool tables are almost identical. Only one ball is slightly out of place. When we hit the cue ball, the movement of both sides starts off the same way but after a little while, not very long, the trajectories become completely different. If I want to predict the future path of the cue ball, I can do so, but I need to know in detail the positions of all the balls on the table, and there are many of them! A small uncertainty about a single ball will shatter any hope of determining the future. Here is the classic definition of determinism, by the mathematician Laplace in 1814. We must consider the present state of the universe as the effect of its previous state, and as the cause for what will follow. An intelligence which, at a given moment, would know all the forces by which nature is animated and the position of every object in the universe if indeed it was powerful enough to submit these data to analysis, would embrace in a single formula the movements of the greatest bodies of the universe and those of the lightest atom: nothing would be uncertain for it, and the future as the past, would appear before its eyes. How can we understand the movements of celestial bodies? Watch this computer simulation of a fictional solar system with two Suns and one planet. It's somewhat like our billiard table. The computer can calculate step by step the movement but can it predict the fate of the system? Will the small planet one day collide with one of the Suns? So... we could predict the future if ... we had an infinite intelligence! But this is not the case ... unfortunately ... or maybe ... fortunately. But then, what can Science do if it cannot predict? Well... if it’s willing to be less ambitious, and more modest, it can still make some very useful forecasts. This requires a different view on determinism! We will not try to predict the future position of a ball amidst a cloud of other balls, but rather look for a probability. The purpose of the forecast is no longer to determine the temperature in Paris on a specific day ten years in the future - this would be pointless - but rather to try and predict averages, statistics, such as the number of hurricanes that will cross the Atlantic in one season. Probabilities rather than certainties? this really is a change in perspective! There is a whole world between theory and practice.
[2] CHAOS 2-The Lego race On October 26, 1676 Isaac Newton writes a letter to his great rival Leibniz. He wants to talk about his greatest discovery but at the same time he wants to keep it secret, so he sends a riddle: «6a, 2c, d, ae, 13e, 2f, 7i, 3l, 9n, 4o, 4q, 2r, 4s, 8t, 12v, x». Leibniz did not decode this message but science historians have eventually decrypted the Latin phrase that was hiding in the anagram: «Data aequatione quotcunque fluentes quantitates involvente, fluxiones invenire et vice versa.» In English: «Given an equation in any number of variables, find its derivatives, and vice versa.» This is calculus: a crystal ball that is incredibly effective at predicting the future. It lets you calculate the future movement of a system when you know its present state and the forces that affect it. It is about solving a differential equation and finding solutions. Let’s look at this in the world of lego people. After all, the mathematical world is a bit like a game, where everything is simpler than in real life; a world that is sometimes childish. The footprints indicate the paths followed by the athletes, their trajectories. Suppose that our little men take steps at a steady pace but that these steps are sometimes long and sometimes short. When the Lego goes fast, two consecutive footsteps are far apart. When they go slowly they are close together. The arrow joining two consecutive steps indicates the speed: it is the difference between two positions at two consecutive moments, hence the word "differential calculus". The hundred meters race. Who will win? 9'58 ", great! Now, movement rarely consists of jerky steps. Look at these moving racecars: They roll, they don’t take steps! What do we then mean by speed? One idea is to say that after all this film consists of 25 frames per second so that one can indeed think that the motorbike is taking 25 "steps" every second! and we can talk about its speed, just like we were talking about the speed of the lego runners. Newton approximates any continuous movement as a sequence of stepwise movements, but with steps that become so small that they become invisible, like in the movies. Calculating the speed of a motion, is called calculating a derivative. It is the goal of "differential calculus". Here are some movements. These arrows indicate the speed, and mathematicians call them vectors. Now imagine the opposite problem. You see arrows drawn across the floor. We call this a vector field. Imagine a field of wheat ... but instead of wheat stalks you have a vector. The lego people’s mission is to move with a speed indicated by the vector field. Easy you say! They look under their feet, they see a vector which tells them their speed, and then they set off in that direction at this speed. A brief moment later they have arrived at a new point with a new speed, then off they go in the new direction and they do this over and over again Walking is not difficult Just put one foot in front of the other and do it again ! " Actually, we should explain what we mean by "a brief moment later!" Newton’s answer would be «An infinitely brief moment! ..» We already saw that a continuous movement is not the same thing as a succession of steps: it’s what you get, when the steps become smaller and smaller. So rather than hopping like a lego person, take a car that rides continuously. You will follow what is called a trajectory; a curve that is tangent to the vector field everywhere. Here you have a vector field in the plane, and two points that are the initial positions of two motorcycles. The theorem of Cauchy-Lipschitz summarizes the concept of determinism: it claims that these points determine the future trajectories. Starting from each point, there is a unique trajectory whose initial position is the given point. Each point has its destiny, different for everyone ... The lego man facing his fate! All he can do is follow his trajectory, and two trajectories can never cross each other. Determining the trajectory from the knowledge of the velocity field is the work of integral calculus, which thus goes in the opposite direction as differential calculus ... And here is a group of lego people, neatly ordered like little soldiers, ready to start walking. Off they go. You see that the nice original order gets broken. It is sometimes said that one determines the flow of a field, as if these characters were floating on a river, each following their course in the current. Think of the flow of humanity, these seven billion lego people moving all over the Earth, or think of the flow of billions of billions of billions of molecules in the Earth's atmosphere. Here is a simple example, almost naive, which will show us a weakness of determinism ... Watch this vector field. The figures move forward and as you can see, those on the left of the central line turn to the left and those on the right turn to the right. In a way determinism is valid: everyone follows his destiny over which he has no control. But on the other hand, two men very close to each other have very different destinies. A little thing can completely change the future. In his little book "Matter and Motion", published in 1876, the physicist Maxwell stresses the sensitivity of physical phenomena to initial conditions. Here is what he writes: «There is a maxim…that the same causes will always produce the same effects [...] There is another maxim which must not be confounded with the first, which asserts that “like causes produce like effects”. This is only true when small variations in the initial circumstances produce only small variations in the final state of the system. In a great many physical phenomena this condition is satisfied; but there are other cases in which a small initial variation may produce a very great change in the final state of the system. " For instance, a small difference in the speed of the car may cause an accident. This dependence of the future on initial conditions is only one aspect of chaos! But there are much more complex situations ... Imagine for example a vector field which is no longer drawn on the ground, but in space. This one for instance that you see on a vertical plane moving back and forth. Now our legos don’t walk: they fly their spaceships! At every moment, their speed is determined by the vector field. See what happens to our unfortunate legos! This is much more chaotic! Imagine a lego person as a soothsayer: impossible with such a roller coaster: His predictions would be mere deceptions. Where will he be in an hour? No one knows! If it is difficult to predict the future of a lego imagine predicting the future of a human being!
[3] CHAOS 3- Newton's apple At the age of 17, Newton wonders why an apple falls from an apple tree and the Moon does not fall on the Earth. Until that time people thought that objects on Earth, such as apples, do not follow the same laws as celestial bodies. Comparing the motion of the Moon and an apple, and thinking that there might be universal physical laws, valid on Earth as well as in the skies, was a real revolution. Physics was dominated by the thoughts of Aristotle. Each object has a place of its own, and if we move it from that place, it will do its best to return to it.. Everything around us seeks its natural balance. An apple tends to go down, since it is its nature. The Moon orbits the Earth because it is its nature. Understanding falling objects is not an easy thing. …and by the way, cartoon heroes have some problems with real physics. This is the first law of physics in cartoons: Any object suspended in space remains motionless until it becomes aware of the situation ". According to a cartoonist at Walt Disney: Cartoons follow the laws of physics unless the contrary is funnier." Newton came up with the "universal law of gravitation": " Two bodies attract with a force proportional to each of the masses, and inversely proportional to the square of the distance between them." Where does this force of attraction come from? How can it be explained? How can two objects separated by billions of kilometers attract each other instantaneously? This mystery was not explained until much later, at least in part, by Albert Einstein. All objects, an apple as well as the Moon, are attracted by the Earth: The force of gravity affects everything that surrounds us... Another great idea from Newton is that forces modify speeds. When a speed increases, we say that the object accelerates. So imagine an abstract apple subjected to no other force than its weight. In the vicinity of an apple tree, gravity is approximately constant. When the apple comes loose, its speed is zero. But the force of gravity changes the speed. The apple falls faster and faster and crashes to the ground. Newton goes further; he explains how to calculate a trajectory if we know the forces. A ball rolling on a plane. What will happen to it? Nothing! It will continue to roll at a constant speed. This statement that we find in physics books is true of course except that it does not match what we are used to in daily life. A billiard ball will eventually stop as frictional forces slow it down. But let’s neglect these forces! When the ball is in contact with the ground, the ground reacts and applies a force in the opposite direction that prevents the ball from piercing the ground... ..so that the total force acting on the ball is zero. If there is no force, there is not change in speed and so the speed remains constant as we have seen before.. But when the ground gives way, the ball is subjected to the force of gravity, its weight, which now changes the speed as there is no ground anymore to counterbalance the weight. Newton expressed all of this in one of the most important formulas of physics: F=m.a F is force, m the mass of the object and a is the acceleration. The acceleration is the rate at which the speed changes, in other words, the derivative of the speed... In our situation, the force of gravity is constant, directed downwards. So the acceleration is constant and directed downward. The velocity vector increases each second by a constant vector. Look at the trajectory: it’s a parabola... If we know the forces that are applied to a system, we know the acceleration. If we also know the initial position and the initial speed, we can then predict the future movement. If you aim very well and if you choose the right initial conditions, you can get incredible motions. An initial condition is not just an initial position but also an initial speed. Look here. Three planets, no Sun, follow three closed trajectories. The motion is periodic. Here things are more interesting. The three planets share the same trajectory! And here, the three planets also follow the same trajectory, a figure of eight. This is planetary choreography... It must be said that such choreographies are real feats: in order to make them work, we have to aim very well, as in billiards, If we change the initial conditions just a little bit, wham!, the beautiful choreography turns to chaos. By the way, why doesn’t the Moon fall on the Earth when it is attracted by the Earth, just like an apple? Newton's answer is amazing: It does fall! If we observe what happens in the vicinity of the Earth, the force of gravity is not constant. It is always directed towards the center of the Earth and is less and less strong as we move away from it. The Earth attracts the Moon by the gravitational force which influences her trajectory. But as this is combined with a rotation, the Moon is constantly falling but never reaching its goal. If the Moon were not subjected to any force, she would move in a straight line pretending not to know the Earth. So let's take a fictitious Moon at 384 000 km above the Earth and let it loose. She falls! Second attempt: let’s launch the same moon with a small initial horizontal speed. The trajectory is deviates slightly but we have not avoided the catastrophe With a higher initial speed the Moon misses the Earth narrowly and continues turning around it periodically A higher speed yet and the moon rotates on a circular orbit (more or less what happens in reality). Too high a speed and good-bye moon! The solar system has eight main planets and thousands of secondary objects, all interacting with one another. In the study of such as system, the description of the initial conditions will involve many positions and many speeds. And just imagine all the data necessary to describe the motion of the atmosphere and its countless molecules. Studying the trajectories of a vector field that depends on a large number of variables, this is the real challenge launched by the great Newton...
[4] CHAOS 4- The swing Simple things first... In the sixteenth century Galileo observed a swinging lamp, and measured the oscillation period by taking his own pulse. How can we describe a swinging pendulum? The tip of the pendulum moves on the blue circle Let’s place this circle on the cylinder, like this Every point of the circle represents a position of the pendulum. The red point for instance is the position at the bottom. Its speed, the red vector, is simply described by a number, positive if it rotates in one direction and negative in the other. We can represent this number in the cylinder’s vertical position. So a single point on the cylinder describes both the position and the speed. The first coordinate, on the base circle, tells us where the swing is and the second, vertical, tells us how fast it’s swinging. So now, let’s put gravity to work and watch the swing! For now we assume that there is no friction at all! The swinging will never end. If the initial position is low and if the initial velocity is not too large, the pendulum moves periodically. We all know that. If we launch it faster perhaps too fast, it goes all the way round! It is still a periodic movement but not in the same way. Look at the trajectory on the cylinder. It follows a vector field... Now observe the more realistic situation where we assume that there is friction. The oscillation slowly decreases, and then finally the pendulum tends to stop. Remember Aristotle's theory: the swing is now back in its natural position at the bottom! Observe the corresponding vector field on the cylinder... Trajectories eventually fall down, to the resting position. This equilibrium position is called a stable attractor. A swing with friction is not so much fun because it eventually stops So we have to push the swing to make it go really high. This is also the point of view of the gentleman represented by the painter Fragonard who seems interested in... the swing! Let’s imagine that the pendulum, in addition to gravity and friction, is also subjected to a pushing force, transmitted by a small nozzle. A modern pendulum with a small jet engine! For example let’s push, but only if the speed and the inclination angle are lower than a certain value. If the swing starts off slowly, it is pushed, it accelerates, accelerates, until the pushing stops as it goes too fast. Friction takes over and it slows down, slows down, until the push comes to the rescue again. etc.. The swing finally stabilizes on a periodic regime. Poincaré talks about a limit cycle. The opposite of chaos, in a way. The periodic regime. Everything in sync! Here is what is called the phase portrait of the cylinder. The limit cycle is in red. We see a spiraling trajectory that gradually becomes the periodic trajectory. Here is simple example that illustrates all of this in ecology: the Lotka-Volterra model, dating from the 1930s. Two populations share a territory. These are for instance rabbits and foxes... We will use our fantasy and take ducks and water lilies. Imagine that the ducks eat the water lilies. When there aren’t many ducks then they eat only few lilies. The water lily population therefore grows rapidly. When there are few ducks and many water lilies, the ducks are well fed and their number increases. But a lot of ducks eat a lot of water lilies! When there are fewer and fewer water lilies the ducks have nothing to eat and their number decreases. We have come full circle and can start again! We can show the situation at a given point in time by a point on a plane: the first coordinate represents the number of water lilies and the second the number of ducks. In fact we get a vector field in the plane... Over time, the ducks and lillies follow a trajectory of this field and reach a limit cycle. The populations of ducks and water lilies eventually oscillate periodically. The belief that any motion, perhaps after a short transition period, eventually stabilizes, either by stopping, or by oscillating periodically, has dominated science for a very long time. One of the first theorems in the theory of dynamical systems, by Poincaré in the late nineteenth century, seemed to justify this. It is about vector fields in the plane. Imagine a vector field like this: In fact, we don’t know the vector field everywhere, but we know how it behaves close to this circle. Take a trajectory that enters the disk. Where can it go? Well, The Poincaré-Bendixson theorem says that there are two possible cases: The trajectory will either get very close to an equilibrium position: as we see here... this is what we saw for the damped pendulum - or it must approach a limit cycle. How can we prove such a theorem? Here is the main idea. Take a point on the circle and observe its trajectory that enters the circle.. Stop! Let’s stop at a point P! Look at the trajectories in the vicinity of P. Is it possible that the trajectory continues and returns the vicinity of P? Let’s suppose that it indeed returns to a point Q. The arc of the trajectory between P and Q, followed by the line QP forms a closed curve. This is the boundary of a certain area, drawn in blue. We can see that a trajectory that starts from Q enters the blue area and then can’t get out anymore! It’s trapped... In order to exit, it should cross the arc PQ but two trajectories cannot cross. Why? Well if two trajectories passed through the same point, this would contradict the Cauchy – Lipschitz theorem: there is only a single trajectory through any given point! The trajectory starting from Q cannot escape through the line QP either as there the vector field is coming in, not going out. You can see that the trajectory from P may well come back not too far from P but then it is condemned to never return... It is said that there is no recurrence: This is the main idea of the Poincaré-Bendixson theorem. This theorem marks the beginning of what is called the qualitative theory of dynamical systems. Even if one has only an imperfect knowledge of a vector field, one can often understand the behavior of its trajectories. In the case of the plane, everything goes well: the trajectories become periodic in time or they approach a position of equilibrium. But Poincaré soon discovered that his theorem is valid only for fields in two dimensions, that is to say for very small systems... From three dimensions on up, we will see that the situation can be much, much, much richer... and pretty! No more nice limit cycles, welcome to the world of chaos!

[5] CHAOS 5- Duhem's bull Ah! … understanding the movement of celestial objects! It’s an old dream... Some people think they can read their destiny in the sky. Can we predict collisions between planets? Could gravitation eject some planets to infinity? Or should we rather expect a stable movement of the solar system? These are delicate questions! As always in mathematics, when faced with a problem that is too difficult, it is better to first look for a simpler situation. Look at this parabolic bowl. If we launch a ball it is subject to its weight and the reaction force which compels it to remain on the surface. We see that the problem is analogous to that of a planet attracted by a sun. The movement seems too regular to resemble a complex solar system with many planets. Take a bowl that is a bit more complicated. The ball is still subject to its weight and to the reaction force of the surface. The movement is really complicated. Let’s take away the weight of the ball but keep the reaction force of the surface. Listen to Pierre Duhem, the philosopher of science, presenting the work of mathematician Hadamard, published in 1898 in an article entitled "On geodesic surfaces with opposite curvatures." A material mass slides on a surface; no gravity, [...] no friction hinders its movement. It describes a line that geometers call a geodesic curve of the surface under consideration. When we choose the initial position of our material point and the direction of its initial velocity, the geodesic is completely defined. Imagine the forehead of a bull, with protrusions from which the horns extend [...] and passages between these protrusions [...] but’s let’s extend the horns towards infinity; we will then have one of the surfaces that we want to study. Extend the horns towards infinity? That must be a mathematician’s idea! Launching a ball on the forehead of a bull with infinitely long horns… We will first use a different example, but one that is in the end quite close to Hadamard’s geodesics: the game of billiards. Here is a rectangular billiard table. I shoot a ball… …and then another, almost in the same way. Too simple... The two balls follow trajectories that are very close. An elliptical billiard table? Still too simple! Here also, two balls that leave with similar initial conditions have trajectories that remain close together. It is as if we were drawing geodesics on the forehead of a bull without horns! Let’s add a ‘horn’: a circular stud. Now look at two nearby trajectories. They hit the stud and the first goes off in one direction, and the other in a completely different direction… Their futures quickly become very different. Here is an infinite plane where balls can roll without any friction. There are three circular obstacles: let’s call them A, B and C. Now here it’s as if we were observing the geodesics on a surface with three horns. Let’s take a ball and shoot it! If we aim well, the trajectory can hit A then B then C then A then B then C etc... a periodic triangular trajectory. Are there other periodic orbits? Hadamard proves a beautiful theorem. He says that if you choose any word written with the three letters ABC with the condition that consecutive letters are different, for example ABABCABC …then there is a unique periodic trajectory that visits the studs successively in the order dictated by the word in question. You see the complexity of the situation: for each word, there is a periodic trajectory. And there are many words! For example, the word ABCBCBCBCBC is a trajectory that pretends to hit only B and C, but once every 11 rebounds it will hit stud A! Of course, we must aim precisely, very precisely! All this makes one think of the real numbers. Some of them have a periodic decimal notation, and these are the rational numbers. 123/999 for example is 0.123123123… and 2/7 is 0.285714 285714 285714... etc.. Irrational numbers have a decimal notation that is not periodic... pi for example. Our billiard table’s similar. Some trajectories are periodic and are described by a periodic word in the letters A, B, C. Others are non-periodic and are described by an infinite word. Still others visit A, B, C a finite number of times, and then go off to infinity and never return. Here is Duhem’ bull: «First, there are geodesics that close on themselves. There are others that, without ever coming back to their starting point never end up infinitely far away from it; some keep turning around the right horn, and others around the left […]; other, more complicated ones, alternate turns around one horn with turns around the other one, following certain rules [...] On our bull’s head [...] there will be geodesics that will go to infinity, one by climbing the right horn, others by climbing the left horn [...].» Imagine the Moon, that has always been Earth’s companion, suddenly deciding to shoot off towards infinity. «Despite this complication, if one knows with absolute accuracy the initial position of a point on the forehead of this bull, and the direction of the initial velocity, then the geodesic that this point will follow during its movement will be unambiguously fixed. It will be quite different if the initial conditions are not known mathematically, but practically" Appreciate the subtlety: not mathematically but practically! Watch these geodesics that start n almost the same conditions. They follow  almost the same path for a while and then they separate! The green, red and blue geodesics have completely different futures. «lf a point is launched on the surface in question from a position that is given geometrically, with a speed that is given geometrically, then mathematical deduction can determine the trajectory of this point and determine if this trajectory moves away to infinity or not. But, for the physicist,  " this deduction is forever unusable.» So far all of this applies to geodesics or billiard ball trajectories. Can this be applied to everyday life, for instance for the movement of planets? The question of determining my future is still unanswered!
[6] CHAOS 6- Smale in Copacabana In the early 1960s, the young American mathematician Steve Smale was working on the beach of Copacabana when he made a discovery ... He discovered a horseshoe! Not in the sand of course! In fact, the horseshoe is an abstract mathematical object! It's another simple idea that attempts to reduce chaos to its most elementary expression. We must first explain an old idea dating back to Poincaré. Here is a vector field in space. A trajectory that starts on this disc makes a full turn and then revisits the disc then another turn and so on. For each point x of the disc, we can observe the trajectory that starts from that point, and wait until it hits the disc again in a new point T(x). A few seconds later, this new point is again transformed by T, and we get a third point T(T(x)). …and so on: at each turn, the point is transformed by T. Instead of studying the trajectory in space, we study a sequence of points in the disc: it's easier to draw! We replace the dynamics of the field in continuous time by the dynamics of the transformation T in discrete time: 0, 1, 2, 3, etc.. This is often much easier to understand. Here the transformation T is a rotation by one third of a turn. The central trajectory is corresponds to the fixed point of the rotation: the origin. Other trajectories are periodic also, but it takes them three times longer to return to the starting point: the rotation is one third of a turn. Here, the transformation T is a similarity. The trajectories are no longer periodic: On the contrary, they approach the central trajectory while spiraling around it. The central periodic trajectory is stable. And here is an unstable trajectory: The transformation T crushes the kitten along its width and stretches it along its height. The center is a fixed point, so its trajectory is periodic. You see that the trajectories located on the horizontal axis approach the fixed point while those on the vertical axis move away from it. Poor little cat! So let’s go on to the horseshoe ... Watch how this square is deformed over time. The transformation T combines a scaling and a contraction, but also a folding. When the square arrives back on the original disk, it has taken the form of a horseshoe. Now watch the behavior of the square as we go back in time, by the inverse transformation. A scaling, a contraction, a folding, but all in the other direction. Let’s try to envision the future and the past of the square. After one round, as we have seen, it became a horseshoe. After one turn the other way, towards the past, it becomes a horseshoe, turned by a quarter turn. The two small vertical stripes you see, at the intersection of the square and the horseshoe are also compressed, scaled and folded. The behavior in the past is the same. And this whole structure is repeated an infinite number of times. A complicated horseshoe! How can we understand its dynamics? Let’s call this area A. and this one B . Well, the amazing thing is that we get almost the same results as Hadamard’s geodesics and billiards : For each finite sequence of A’s and B’s, even when allowing repetitions, for instance BABB, there is a periodic point that will follow this route exactly. In the example, it is a point of B that jumps to A, then jumps to B and then jumps to B again , and then jumps back to the starting point. Here is the only point of A that always remains in A: it is a fixed point This fixed point is in B. Here is one that alternates between A and B. But, as with the billiards, this is not limited to finite sequences. Give me an infinite sequence of your choice BBAABBBAABABA etc.. etc.. Well, there is a point following exactly this route in its future ... When you think about it, this is incredible: every possible future is true for at least one person… (Oops, I meant for at least one point!) Prediction is impossible because everything is possible. Free will rediscovered! Anything is possible, there’s a fine slogan for chaos! Thank you Mr. Smale. Smale realizes that it is not always necessary to know things exactly in order to understand them. A distorted picture is often enough to recognize a face. He shows that the horseshoe is stable. Mind you, this does not mean that the trajectories are stable: on the contrary, the trajectories have a big sensitivity to initial conditions. What we mean is structural stability. Distort Mr. Smale’s picture, and you will still recognize him. On the right we have the original horseshoe. On the left another horseshoe, slightly modified, is being formed. After all, when you stretch the square into a rectangle and then fold it in the form of a horseshoe, you can do it in several ways. This is somewhat analogous to a situation where we would study two solar systems where the planets have slightly different masses Two almost identical horseshoes. We can do exactly the same as we did with the original horseshoe and draw areas A 'and B' the same way. And it can be shown in the same way that the trajectories of this new horseshoe are still described by sequences of A’s and B’s . The modified horseshoe is just as chaotic as the original one! Chaos is really there, indestructible. In fact, Smale shows that the movements on the two horseshoes are somehow "identical". Look, the two trajectories, right and left follow the same choreography, they visit the same A, A 'and the same B, B'. We can match a trajectory of the second horseshoe to any trajectory on the first and vice versa. It is in this sense that the structure is stable: we can deform the horseshoe any way we want, not only does it remain chaotic but it keeps the same dynamics as before. Individual trajectories are unstable but the dynamics as a whole are stable. Early in the twentieth century Poincaré, with characteristic flair, explains how it is possible to understand a dynamic even if one does not know it very well. «You ask me to predict the phenomena that will occur. If I had the misfortune to know the laws of these phenomena, I could only get there by intractable calculations and I would not be able to give you an answer; but, as I am so lucky as to be ignorant of them, I'll answer right away.. And what is most extraordinary, is that my answer will be correct.» The coexistence of chaos, and hence the instability of individual trajectories, and structural stability, a global property, is absolutely remarkable. I'm unstable, but the world around me is stable! Reassuring ;-)
[7] CHAOS 7- The Lorenz butterfly When we look at the movement of the atmosphere we soon realize that it is infinitely more complex than that of the solar system. The atmosphere is a fluid that has, at each altitude above each point of the Earth’s surface, a speed, a density, a pressure, a temperature etc.. All this varies over time. It is of course unthinkable to know all this infinite number of data. It's almost as if we were in a space with an infinite number of dimensions: To understand something about it, we must make approximations. In 1963, Edward Lorenz simplified, simplified, and simplified the problem again. He simplified it to such a degree that there is no guarantee that his equation still has anything to do with reality. His model of the atmosphere was reduced to only three numbers (x, y, z) and the evolution of the atmosphere to a tiny equation: Each point (x, y, z) in space symbolizes a state of the atmosphere and the evolution follows a vector field. For example, but this is only an example, the first coordinate could represent a temperature, the second the wind speed and the third the humidity. Here, it is cold, the wind blows and it rains ... Here, the opposite is true ... When we follow a trajectory of the field, we are following the evolution of the weather ... The forecaster just needs to solve a differential equation! This is what Lorenz saw when he studied his model. Does all of this have anything to do with real weather? Not clear at all! It is what physicists often call a toy model, that is used to try and understand the broad outlines of a some complex behavior. In fact, Lorenz had only these sorts of graphs to look at because his computer, in 1963, was quite primitive. Let’s look at two atmospheres, represented by the centers of these two balls that are extremely close together, so that they are almost identical, and let’s observe what happens to them. At first the two evolutions remain indistinguishable. But then they split up significantly: both atmospheres become completely different. This is chaos : the sensitive dependence on initial conditions. In 1972, Lorenz was going to present his work at a prestigious conference, but he was late in sending the title of his lecture. The organizer was in a hurry to send the program to the participants, and so he chose the title: “Predictability : Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” The butterfly effect was born! Mathematical concepts are not often well understood by the general public The image of a small, frail butterfly having an influence on the world is very poetic ! The idea was very successful and was modified. There is always a butterfly, but sometimes it comes from Africa, and sometimes from China. ..and it is responsible for disasters in New York…or Chicago. Interesting... The idea found its way into literature, music and movies. The list of films based upon it is endless. A lot of emotions in Babel, the 2006 film by Alejandro González Iñárritu The insignificant incident of a gunshot in Morocco will change the lives of an American couple, a Mexican nursemaid and a Japanese girl. An interplay of small causes and large consequences. ..how chance determines the fate of humans. But unfortunately, only one half of Lorenz’ message made its way to the general public. Can chaos theory be limited to the statement that it is impossible to predict the future in practice? How can a scientist resign himself to such an admission of failure? Lorenz’ message is a lot richer. Here are two trajectories of the Lorenz system, one blue and one yellow, that do not necessarily take off from close initial conditions . Let’s erase, say, the first ten seconds, of the movement and let’s observe what follows…. What do we see? That the trajectories are indeed very different, they seem a little crazy and quite unpredictable. But they accumulate on the same butterfly-shaped object that does not seem to depend on the initial position. They seem to be attracted to this butterfly. That is why we speak of the Lorenz attractor. A strange attractor! Now this is a positive scientific phenomenon, not as famous as the butterfly effect. Instead of observing just two trajectories, let’s look at a lot more. Look at all these balls, each one of them representing a simplified atmosphere. After a while, they all accumulate on the same butterfly. A very nice object, that one can admire endlessly. These are real problems for mathematicians and scientists in general. Instead of describing the future starting fgrom a given initial condition, which we know is futile, we will try to describe the attractor. What does it look like? How do the internal dynamics work? In the 1970s, Birman Guckenheimer and Williams proposed a simple model for trying to understand the Lorenz attractor. Here is a strip of paper, Let’s fold it, and cut it …and glue it together. ..and we have a special object in space! Let go to the starting line. Here we go, we drive and little while later, we're back on the starting line ... But we find ourselves on another point from which a new path starts. The position on the starting line is given by a number x between 0 and 1. At zero we are on the left, at 0.5 in the middle and at 1 on the right. If we start from the point x, then the return point is 2x if x is smaller than one half, and 2x-1 if it is larger than one half. In other words, when the car moves, the points where it meets the starting line are doubled at each turn, except that we need to subtract 1 if the result is greater than 1. It's a bit like the horseshoe : the dynamics in continuous time got replaced by dynamics in discrete time, namely the successive passages on the starting line. Look, we start off from 1/3, we arrive in 2/3, then 4/3 but we must subtract 1, that is to say 1/3: We are back on the starting point after two rounds. So we have a periodic trajectory of period 2. Here is a trajectory of period 18. A periodic trajectory passes successively through the left and right ears following a certain sequence. Well, it can be shown that all sequences are possible. And of course, not all trajectories are periodic. Whatever the infinite sequence of left and right, there is a trajectory that follows this destiny! What is the relationship between the Lorenz model and that made of strips of paper? Well, it was’nt until 2001 that the mathematician Tucker showed that the paper strip model accurately describes the movement on the Lorenz attractor. For each trajectory in the Lorenz attractor, there is a trajectory on the paper model that behaves in exactly the same way. Can understanding the movement of the atmosphere then be reduced to continually multiplying by 2 a number between 0 and 1, and subtracting 1 if the result is greater than 1? Of course not! All this is too simplistic, but it is an illustration of the phenomenon. Simple things? Mathematicians love them! Why has the butterfly effect become so popular? Perhaps because it gives us back our personal freedom. The legacy of Newton’s cold determinism could lead to a kind of fatalism. The Lorenz butterfly claims, small as we are, that we can have an influence on the world! Good news for us!
[8] CHAOS 8- The Lorenz waterwheel Listen to Ed Lorenz telling us about his concept of chaos in three points: «lf a single flap of a butterfly’s wings can be instrumental in causing a tornado, so also can all the previous and subsequent flaps of its wings, as can the flap of the millions of other butterflies, not to mention the activities of innumerable much more powerful creatures, including our own species? » (We have already seen that the general public understands this first point well) If a flap of a butterfly's wings can be instrumental in causing a tornado, it can equally well be instrumental in preventing it.» Yes! But the third point is the most important and this one provides work for scientists. «More generally, I am proposing that over the years, minuscule disturbances neither increase nor decrease the frequency of occurrence of various weather events such as tornadoes. The most they may do is to modify the sequence in which these events occur." Let’s return to the Lorenz model. Remember that each point in space represents a state of the atmosphere. In some areas the weather is fine and in others a hurricane blows! Suppose that the area corresponding to a hurricane is this little ball. Let’s take an initial climate condition and let’s observe the trajectory. Occasionally, a hurricane occurs, when the trajectory enters the ball. Let’s measure the proportion of the time between zero and T during which a hurricane rages. It seems difficult to guess at which specific moments the trajectory enters the ball, but the average time inside the ball converges to a limit when T becomes infinitely long. Here it is 5.1%. We now take another initial condition and do the same calculation. Well, again the trajectory spends a certain amount of its time inside the ball but Oh! How about that, it's the same as before: 5.1%! Lets’ try it with yet another trajectory... It works! We obtain the same limit! …and yet, the trajectories are very different, they are sensitive to initial conditions on the wings of the butterfly... We add another ball, pink maybe, and a third one, green... Perhaps periods of heat waves or snow? When a trajectory develops, it traverses the balls in a certain order. To see what happens, let’s take three trajectories starting from three initial conditions. Look. On the trajectories, the situations hurricane / snow / heat wave alternate incomprehensibly, and in a different way for every trajectory. Impossible to understand. But the proportions of yellow, green and pink become the same for all of them Here they are, 5.10, 14.03 and 7.40% It is as if we picked yellow, green or pink balls at random with well-defined probabilities. Listen to Lorenz again: «Small changes do not increase or diminish the frequency of weather events like tornadoes. The only thing they can do is to change the order in which these events occur." This is a scientific statement! The purpose of the forecast has changed. We now try to predict averages, statistics, probabilities. The idea is that these statistics are - perhaps... insensitive to initial conditions and to the flapping of the wings of Brazilian butterflies! This hypothesis - unproven... the possible coexistence of a meteorological chaos where the future movements are sensitive to initial conditions, and a statistical stability, insensitive to initial conditions, took a long time to be formulated mathematically. Lorenz was probably a little embarrassed by the simplistic (to say the least!) theoretical side of his 3-parameter atmosphere. With the help of two physicists, Howard and Markus, he developed a "real physical system", even if it is still simplistic and far removed from the true weather phenomena! Here is a mill. It consists of a wheel, with buckets suspended all around it. The buckets have a hole in the bottom of course, and the water can run out, all the more quickly if the water level in a bucket is higher.. We open a valve at the top and watch the movement. The mill turns sometimes to the right, sometimes to the left and we seem to be totally incapable of predicting which way it will turn in a few seconds. The movement seems totally erratic, unpredictable, chaotic. In fact, is there a relationship between the mill and the Lorenz attractor? Let’s choose three numbers to describe the mill, for instance the angular velocity and the two coordinates of its center of gravity. The evolution of these numbers traces a yellow curve in space. Amazing is it not? Our mill is moving ...like a butterfly! Let’s change the initial position imperceptibly. All buckets are initially empty and the left wheel is turned by 2 degrees, while the right wheel is turned only 1.9996 degrees. (we would need a good microscope to notice the difference). Yes, there clearly is a "sensitive dependence on initial conditions"! After a while, the two mills have completely different behaviors. Let's see if Lorenz’ statistical statement is true for the mill. Our two mills take off from almost the same position. Let’s measure their speed: for instance 25 times per second during 5000 seconds, so 125 000 observations. At each observation, we note the rotational speed of the wheel. A simple idea is to do what statisticians do: a bar chart to illustrate the distribution. We have 35 equal intervals for the speed and we count the number of times that the measured speed falls in each interval. Here is the result. Some intervals appear to be visited more than others. Our two mills behave in very different ways. There is sensitive dependence on initial conditions. But the observation is that the two sets of data, although different, are statistically identical. The bar charts tend to become identical. It works! Lorenz seems to be right! When the statistics of a trajectory’s future are insensitive to initial conditions, we say that the dynamics have a Sinai-Ruelle-Bowen measure, or SRB measure. The goal of a forecaster is then to determine these statistics. For example, let’s observe the evolution of temperature over time, one of the coordinates in Lorenz’s equation. We draw another bar chart, as we did with the mill. For each temperature range, we note the proportion of observations that fall within this range. Even if we consider trajectories that are quite different from one another, after a while the diagrams tend to become identical. The Lorenz attractor has an SRB measure. It is as if the temperature was changing at random but with very specific probabilities. But there is a lot left to do mathematically and physically. We must find these probabilities and these distributions in order to be able to say something useful! Let’s hope we have done justice to Lorenz who did not say merely that the future depends heavily on the present. Many people before him had said that. But his contribution is also, and perhaps most importantly, to show that by refocusing on statistical issues, Science can still make predictions.
[9] CHAOS 9-Chaos research today There are many kinds of dynamics. Some are complicated, others are not. Watch this vector field in space that depends on a parameter a. Let’s start by choosing a = 0.3 and let’s look at what happens for a large number of initial conditions. After a while, they all fall on a periodic trajectory. No chaos there. There is an SRB measure of course as everyone eventually turns around on a closed curve. Let’s change the parameter slowly. Look, for a = 0.335 the periodic trajectory splits up... Still no chaos, but a periodic trajectory that is twice as long... And then for a = 0.380, the trajectory splits up again and is now fourfold. Then things accelerate. For a = 0.405 things become complicated and chaotic. But what is surprising is that if we continue to increase the parameter a, sometimes, without warning, the chaotic dynamics become simple and falls back to a periodic orbit. How can we understand these bifurcations? What is the most common behavior in nature? Chaotic? Non-chaotic? It is not clear at all. For centuries we had no idea that chaos could exist... and today we see it everywhere! For each value of a, we can mark the trace of the attractor on a plane and we draw it in red. As the parameter varies we obtain this drawing that looks like a piece of lace, and this is called a bifurcation diagram. Nice... but not easy to understand! Here are some more dynamics. Mathematicians seek to establish results that are always valid, but they often start by studying examples. They then hope that what they see in a simple example can be realized generally, for all cases. Remember the idea of Lorenz, Sinai-Ruelle-Bowen, SRB. The proportion of time that a trajectory spends in a ball converges to a limit that doesn’t depend on the initial condition. Is it reasonable to expect that this is ALWAYS true? That would be too nice... So unfortunately the answer is no, as we will see on a small example discovered by Bowen. Watch this vector field in the plane. There are three equilibrium positions. A trajectory starts... and after a while, it approaches an equilibrium position and then moves on to another one. When it approaches an equilibrium, it slows down and remains in its vicinity for some time. Then it restarts towards the other equilibrium and remains there even longer. Back to the first equilibrium where there is an even longer delay…etc…etc So here is a little green disc. Let’s observe the transit time in this disc. Well, the trajectory stays there for long periods so that the proportion of the time is close to 100%. Then we leave the disc for a very, very long time. The proportion falls to a value close to zero, then it goes back to nearly 100% and then falls back and so forth. You see, there is no convergence. There is no Sinai-Ruelle-Bowen measure! So what can we do? Abandon this idea? Say that Lorenz was mistaken? Well no! It should be understood that the example we have just seen is very special. If we change the vector field just a little, like this: …there is no longer a problem with the new field. The new trajectories travel in the plane and eventually approach a periodic trajectory, so that statistically everything behaves periodically and now we do have a Sinai-Ruelle-Bowen measure. So the question is whether these measures show up, not for ALL of dynamical systems, But here again, we must temper our enthusiasm... Watch this vector field. Here is a trajectory. All is well, it accumulates on a butterfly, as we have seen before... If the initial condition moves slowly on the blue axis, everything continues as it should: the trajectory continues to accumulate on the same attractor, even if it takes a little while to get there. And we could easily check that the statistics haven’t changed. But all of a sudden, without warning, surprise... the trajectories accumulate elsewhere… on another attractor that has nothing to do with the first! Space appears to be split into two regions: if one starts with a point in the first region, the trajectory accumulates on the first attractor, and if we start in the second region we end up on the other... In other words, we cannot say that there is an SRB measure since the long-term behavior depends on the initial condition. There are two SRB measures! For certain initial conditions, statistics follow the first and for others the second! In the 1990’s, the Brazilian mathematician Jacob Palis formulated a whole set of problems that, if resolved, would allow a global vision of chaos. One of them is to claim that the situation we just encountered should be typical. In all dimensions a typical vector field should have the property that it has only a finite number of attractors. A typical initial condition should be drawn to one of these attractors. Each attractor should have a Sinai-Ruelle-Bowen measure that describes the asymptotic statistics of the typical trajectories that it attracts. A whole group of mathematicians is hard at work on these conjectures, and they are being solved methodically, bit by bit. A global picture seems to be emerging. Is this picture is too optimistic? Time will tell. Today we no longer think of determinism as the evolution of an individual trajectory, but rather as the collective evolution of a whole set. Sensitivity of trajectories to initial conditions is compensated by a kind of statistical stability of the whole set. Did Buffon, in 1783 not foresee this when he wrote this magnificent sentence about a complex and chaotic world that on has to try to understand globally? "Everything happens, because with time everything meets, and in the free range of spaces and in the continuous succession of movement, all matter is stirred, any form given, all figures printed; so everything is coming or going, all is joined or running away, all is combined or opposed, everything happens or is destroyed by relative or contrary forces, that are the only constants, and balancing without harm, they animate the universe and make it into a theater with ever new scenes and objects incessantly reborn ".

Subtitles of Dimensions [1] Dimension 2 My name is Hipparchus I lived in the second century before the birth of Christ, and I don't think I'd be bragging if I told you that I am the father of the sciences of Geography and Astronomy. You know, I wrote more than 14 books but unfortunately they have almost all been lost in the mists of time. I was responsible for the first catalogue of the stars, founded the field of mathematics called trigonometry and even invented the astrolabe. Fortunately, my brilliant successor Ptolemy, three centuries after my time inspired by my work, took up where I left off, and nowadays historians sometimes can't determine what was my contribution and what was his. Ptolemy's manuscript the "Almagest" was the first scientific treatise on astronomy and his book "Geography" contains the first map of the known world. Geography and geometry both deal with the study of the Earth Geography is concerned with making visual representations of the Earth whilst geometry is concerned with measuring it. The shape of the Earth is roughly spherical. Let's forget for the moment that it's slightly flattened at the poles and pretend that it really is a perfect sphere. You probably know too that all the points of the sphere are at the same distance from its center. The arrow that you can see now, starting at the center of the sphere and ending at a point moving on the surface, has a constant length. Let's choose an axis for our sphere: a line through the center. When we cut the sphere along a plane that contains this axis we carve out a great circle which divides the sphere into two hemispheres. If we chop the sphere up using some sort of guillotine that slices down this axis we trace out the meridians. These are half circles going from the north pole to the south pole of the Earth. And now if we slice the sphere up along a plane at right angles to the axis we get a bunch of circles called parallels. So, now our sphere is covered by two networks of curves the meridians and the parallels One of these parallels should be very familiar. It's the equator, half-way between the two poles. For historical reasons one of these meridians was chosen to be the principal meridian, it's the one passing through the Greenwich Observatory in England, To specify the position of a point on the Earth's surface we can start at the point where the Greenwich meridian meets the Equator, and walk round the equator a distance measured by an angle called the longitude - colored red then you go up along a meridian some way measured by an angle called the latitude - colored green, finally arriving at our desired destination. Any point on the Earth is precisely described by just these two numbers: its latitude and its longitude. Since we need two numbers to specify a location on the surface of the Earth we say that the sphere is 2-dimensional. and mathematicians often call it S². Finally, if we let our little plane leave the Earth and fly off into space then to locate it we need to give three numbers latitude, longitude and... the altitude above the Earth. Since we now need three numbers, to say where we are in outer space we say that space is 3-dimensional. Look at the paintings on the wall, there's a portrait of Ptolemy -- the father of map making. How do we draw the Earth? One method is to project it on a plane. Let's choose a city, Dakar for example, We draw a straight line from the north pole through Dakar Our line hits the table at some point that we call its projection onto the table. Any point on the Earth's surface can be projected onto the table in this way. The closer our town is to the north pole the further away its projection on the table is. In fact it can even end up off the table. For this reason we say that the north pole doesn't have a projection or, more correctly, that its projection is at infinity. The whole Earth, with the exception of the north pole, can be represented on the plane of the table. This map of the world is called stereographic projection. Of course, our stereographic projection doesn't preserve sizes. South America appears tiny compared to North America. To get a better idea of what this projection does, we'll roll the Earth along just like a giant ball, and will always project from the highest point. The projections of the continents waltz around in the plane, taking turns at becoming bigger and smaller. But if we take a closer look, we see that shapes don't change even if lengths do. For this reason we say that stereographic projection is conformal. What happens to the meridians and the parallels under the projection? When we project from the north pole, the meridians become radii emanating from the south pole and the parallels, concentric circles. And as the Earth turns, you see that both the meridians and parallels. always project to either circles or straight lines. Stereographic projection transforms circles drawn on the sphere into circles drawn on the plane, except for those circles passing through the pole from which we project, whose projections are in fact straight lines in the plane. Now here's our rolling Earth from below. From this point of view we see the meridians and parallels form two bundles of circles. All of the meridians converge at two points, the north and the south pole. Do you recognize this one here? Yes, it's the Greenwich Meridian, the end of the first stage of our journey towards the fourth dimension.
[2] Dimension 3 Now it's my turn to show you round the geometry garden My name is Escher and I was a Dutch artist in the twentieth century. Geometry was a constant source of inspiration for me I'm a past master in the art of drawing fantastic tilings. Look at this self-portrait in a spherical mirror. One of my most famous drawings shows lizards, drawn on a plane, that manage to break out of the paper. Now, perched on high, they contemplate their previous existence as flat life. To prepare ourselves for life in four dimensions, we are going to use the ideas behind both my engraving and a little book published at the end of the 19th century by Edwin Abbott, an English clergyman, called “Flatland”. Let's try to explain to these flat beings, these creatures living trapped forever in a plane, what our everyday life in three dimensions is like. Let's imagine that one of these lizards manages to escape his miserable existence for a moment and climbs out and up onto a viewpoint looking down on his world. How would he explain to his fellow reptiles the existence of objects in three dimensions? As a first attempt he could try to pass some three dimensional objects through his flat world. Here, for example, is a tetrahedron with its 4 faces, passing through the lizards' plane. The flat creatures see a green triangle appear suddenly then gradually shrink away. This is all they see since their senses are entirely restricted and they cannot perceive anything outside of their plane. Each time that a lizard sees these green polygons appear, change shape and disappear again they might imagine the form of the object that has just crossed his plane. To see how hard it is to visualize the form of a geometric body from its cross sections try to guess what is crossing through the plane now! A tetrahedron. And now? It was a cube ! Of course you have to remember that these lizards don't have the same perspective that we have. All they see is a sequence of polygons and they'll have to develop an understanding of depth in order to fully appreciate the shape of the body. And now what? An octahedron with its 8 faces And an... icosahedron, it's a solid with 20 faces. And finally... the dodecahedron, 12 faces, 20 vertices and 30 edges... Now we're going to show you some cross sections and only cross sections and you have to guess the polyhedron hiding behind them. That was a tetrahedron That was a cube It's getting hard, isn't it? You see now that these creatures stuck in two dimensions have to develop a good geometric intuition if they want to understand something of the third dimension, that we take for granted. We'll have just the same kind of problems to get a feeling for the fourth dimension. Here's a second method to explain polyhedra to our flat friends. Start by inflating a polyhedron, so that the vertices and the edges are on the surface of a sphere. Then, we stereographically project onto the plane of the lizards, so that our 2-dimensional friends may enjoy the spectacle. Of course, we can spin the sphere around, and with it our tetrahedron, just as we did before with the Earth. Let's take a look at the cube and see how many vertices, edges and faces it has. And now here comes an octahedron. You see the 8 colored faces. Look how the projections of the edges are arcs of circles. Now here comes an icosahedron. Its structure is more complicated but its not hard to understand, even for the lizards. One can see 20 faces, 12 vertices and 30 edges (Can you count them all?) Finally, here's a little geometric jewel -- the dodecahedron. Now for some excercises ! Let's take ourselves down into two dimensions and try to recognize the polyhedra. from their stereographic projections. It's easy isn't it? You can see the 4 faces, 6 edges and 4 vertices. There, it's a tetrahedron. Now what's this one? 6 faces each with 4 edges That's right! It's a cube. That was harder, wasn't it? The faces are triangles. 5 edges start out at each vertex. There are a lot of faces. Perhaps 20? It's an icosahedron. Well done!! Let's look at the dodecahedron. Each face is a pentagon. If we count them there are 12 faces. Three edges start at each vertex. These 5 solids have always fascinated geometers. The Greek philosophers attributed a magical importance to them by associating one of the fundamental elements from which the world is formed to each of them. We call these figures the Platonic solids. So we agree then: It's not easy to get a feeling for the third dimension when you are flat! There is more than one way to do this but our stereographic projection gives a good idea of what's going on . Now let's get ready for the fourth dimension. Prepare your imagination for a workout!
[3] The fourth dimension. My name is Ludwig Schläfli. I am a Swiss geometer. I lived during the nineteenth century and I'm going to open the door to the fourth dimension for you! Even if I say so myself, I was a visionary. I was one of the very first to understand that spaces with many dimensions really exist and that their geometry can be studied. If flat creatures living in a plane can understand 3-dimensional polyhedra, then why shouldn't we understand polyhedra in four dimensions? One of my main achievements was to describe all regular polyhedra in four dimensions. What is the fourth dimension? A lot has been written on the subject; Science fiction writers never tire of talking about it! I'm going to explain things on the blackboard. You'll see that this blackboard has a bit of magic about it. What's important is to prepare yourself to forget about the world which is familiar to us and to imagine a new world that our eyes and our senses have no direct access to. We’ll have to be smart, just like the lizards were before. I'm going to climb up to a viewpoint that, unfortunately, you cannot see and I'll try to describe what I see from there. But before we begin I'll draw a straight line on the board. Let me just mark the origin here. Each point on this line can be located by its distance from the origin, with a minus sign, if it is on the left or a plus sign if it is on the right. Usually the number is denoted by x and is called the abscissa. Since the position of a point on a line can be described by a single number, we say that the line has 1 dimension. Now, I draw a second axis, perpendicular to the first one. Each point in the blackboard plane is now completely described by two numbers, usually denoted x and y : the abscissa and the ordinate. The plane has 2 dimensions. If you had to explain to some being living on a line what it is to be a point in the plane, that is unknown to him, you could simply say "a point in the plane is just a pair of numbers." Let's go to the third dimension. The chalk now writes in the air and draws a third axis, perpendicular to the two previous ones. A point in space is described by three numbers, x, y and z. One could say to the reptiles that are curious to know about our world "A point in space is just three numbers" Let's go to the fourth dimension. One could try to draw a fourth axis perpendicular to the others, but that's impossible! So, we have to do something else instead. Of course, we might just say that a point in the fourth dimension is nothing other than 4 numbers, x,y,z,t. That doesn't help us a lot! In spite of the difficulties, we are going to try to get a feeling for this geometry, As a first attempt at understanding we shall proceed by analogy. Here is a segment... and an equilateral triangle... and finally a regular tetrahedron. Our magical blackboard enables us to draw in space. How can we keep this up in 4 dimensions? Observe that the segment, the triangle and the tetrahedron, have 2, 3 and 4 vertices, respectively. Therefore, we can try to continue with 5 vertices! Let's go then. For the segment, the triangle or the tetrahedron, an edge connects each pair of vertices. So we have to connect the 5 vertices in pairs. We count one edge two, three, four, 5, 6, 7, 8, 9, and 10 edges. In the tetrahedron there is a triangular face for each triple of vertices We proceed the same way, which gives us 2, 3, ..., 10 faces. But, if we keep going, by analogy, we have to add a tetrahedral face for each set of four vertices. There are 5 sets. That's it! We've constructed our 4-dimensional object. We'll call this the "simplex". Let's spin it round in space a little as we did with the tetrahedron. Of course, you have to imagine the simplex spinning in a 4-dimensional space, what you see is only its projection on the blackboard. What makes things a touch complicated is that faces get tangled and that they cross each other. Well, some experience is required to be able to see in four dimensions. We're going to take the simplex, which is in 4D space and move it gradually so that different cross sections of it meet "our" 3-dimensional space. In the same way that reptiles could see a polygon appearing and disappearing, we'll see a 3-dimensional polyhedron which appears, changes shape and then vanishes. Here is the simplex passing through our 3-dimensional space. We're now going to meet more 4-dimensional polyhedra passing through our own 3-dimensional world. Here is the hypercube, a member of the family that starts with the segment and continues up through the square and the cube. I must confess that getting a feeling for the geometry from the slice method like this is rather tricky... I discovered the analogues of the icosahedron and the dodecahedron. They have complicated names but I'll just call them 120 cell and 600 cell since the former has 120 faces and the latter 600. Look at the 120 cell, it's just passing through our space. And now here's the 600 cell. Of course, when I say that a 4-dimensional polyhedron has 600 faces, I mean 3-dimensional faces. Yes, these 600 faces are 600 tetrahedra. As for the 120 cell, it consists of 120 dodecahedra! In a minute, we'll see how we can get to know them better. To observe these 4-dimensional objects with our 3-dimensional eyes, we can look at their shadows. The objects are still in 4D space but they are projected onto our 3D space exactly like a painter might project a landscape onto his canvas. We've already done just this with the simplex. Here is the hypercube. Of course, it's spinning in space so that we can appreciate all the details. Notice for instance that the hypercube has 16 vertices. Here's a little newcomer. It's the most beautiful of my discoveries. An object that I call the 24 cell, it has absolutely no analogue in three dimensions. It's a purely 4-dimensional creature. I am very proud of my discovery. Look how wonderful it is ! 24 vertices, 96 edges, 96 triangles and 24 octahedra. A real little gem! Here is the shadow of the 120 cell in all its majesty! A rather complicated majesty, you have to agree! Let's get inside and have a look at its structure. Look: 600 vertices, 1200 edges. 4 edges start at each vertex. A completely regular structure. All vertices, all edges play the same role. It's a pity that the projection breaks the symmetry. Let's work your imagination a little. Imagine the object in 4D space where a huge group of rotations permutes all these vertices and edges. The champion is... the 600 cell. Like a gigantic macromolecule with its 720 edges and 120 vertices, and 12 edges starting from each vertex. Our exploration of 4-dimensional polyhedra won't stop here as their stereographic projections are bound to give us a better feeling for their geometry.
[4] The fourth dimension...continued. The S² sphere sits in 3-dimensional space, even though it's 2-dimensional. In the same way, we can study the sphere in 4-dimensional space. It contains all the points that are at the same distance from a centerpoint. But now, to determine the position of a point on this sphere, we need three numbers. This means that the sphere has three dimensions, and so we'll call it S³. You won't be able to see this sphere in 4-dimensional space because your space has only three dimensions, and the screen only has two! I can only call upon your imagination. To get a better understanding of 4-dimensional polyhedra we can do just what the lizards did with the 3-dimensional polyhedra, we first inflate them so that they lie on a sphere and then project this stereographically onto the plane. This time, we'll inflate the polyhedron until its faces lie on a hypersphere in 4D space and project stereographically back into our own 3D space. I'm going up to the north pole of the sphere in 4D space and I'll project it down to you in your 3D space. You can't see where I am -- just remember how the lizards couldn't see their kinsman up on his perch either. Now we're in exactly the same situation. Here's the simplex. You can see its 5 vertices and its 10 edges. Of course, in this view, edges are circular arcs. So now we have a situation like that of the 3-dimensional polyhedra projected stereographically onto a plane. Here's the hypercube. It's easy to recognize it from its 32 edges and 16 vertices. Seeing things this way is so much easier than with the shadow method or the 3-dimensional cross-sections. Here's the 24 cell with 24 vertices and 96 edges! Finally, the 120 cell and the 600 cell. Let's add the 2-dimensional faces, to get an even better view! The simplex, with its 10 triangular faces. Of course, these 2-dimensional faces are pieces of spheres, just as before when we saw that the edges were circular arcs. The simplex is spinning in 4D space, before being projected stereographically; remember when the Earth was spinning like a ball and we saw the motion of the continents. Now and again, a face passes through the projection pole and its projection becomes infinite: it looks like it blows up on the screen. Let's take a quick look at the hypercube. You see that space is divided into 8 cube-shaped zones, these are the 3-dimensional faces of the hypercube. As for the 2-dimensional faces, they are squares (though rather bloated and twisted). There are 24 of them. Ah my favorite! The 24 cell. Look at that! The 24 cell's gorgeous! 24 vertices, 96 edges, 96 triangles and 24 octahedra. 8 edges start at each vertex. Here's the 120 cell, let's try to understand its geometry better. 4 edges start at each vertex. The 2-dimensional faces are pentagons. There are 720 of them! These 720 pentagons form 120 dodecahedra. Look at all those dodecahedra fitting nicely together. Amazing, no? Let's finish with the 600 cell with its 600 3-dimensional tetrahedral faces, its 1200 triangular faces its 720 edges and its 120 vertices. You can take my word for it, this object has 14400 symmetries! Well there you are, we're done with our first voyage into the fourth dimension... It's a dimension full of amazing things. Of course, the mathematician's imagination isn't limited to the fourth dimension. There are the fifth, the sixth, the n-th dimension, and even... the infinite dimension! Each dimension has its own character; but if you ask me, the fourth is the prettiest. Why? Maybe because, after all, it has a sort of physical reality. Einstein's relativity theory, dating from the early twentieth century, binds space and time together into a 4D space-time. A point in space-time is an event, characterized by its position in space x,y,z and by the time t when it occurs. Relativistic physics therefore requires an understanding of 4-dimensional geometry. It's interesting to notice that the discovery of this 4-dimensional geometry precedes by some fifty years the discovery of relativity. It's one of the many interactions between mathematics and physics that the history of science delights in.
[5] Complex numbers. I'm Adrien Douady My entire life's work in mathematics was centered on the complex numbers. My contributions helped advance both algebraic geometry and the theory of dynamical systems. Complex numbers have a long history. You see here, on the left, Tartaglia and Cardano, mathematical pioneers who lived during the Renaissance. On the right, Cauchy and Gauss, who consolidated the theory in the nineteenth century. Complex numbers are not really as complicated as the name might lead you to believe! At first they were called "impossible numbers" even today they are still sometimes called "imaginary". Well it's true, it does take a little imagination ... yet today these numbers are everywhere in science and are not really mysterious anymore. In particular, thanks to them, one can construct beautiful fractal sets, something I worked on a lot. I even produced a film "The dynamics of the rabbit", it was one of the first animated films in mathematics. Let me begin by explaining the complex numbers on the blackboard. Mathematicians just love writing with chalk... You'll see in a minute that my ruler, T-square and protractor behave rather oddly sometimes... Let's draw a graduated line on the blackboard. One of the most beautiful ideas in mathematics is to link geometry to algebra. This is the starting point of algebraic geometry. Just as we can add numbers, we can add points. Here is a red point on the line and another blue one Let's add these two points. We get the green point! One plus two equals three! When the red and blue points move, the green point which is their sum must move too. More interesting still is multiplication of points. Let's look at multiplication by -2 for instance. It transforms the point 1 into the point -2, of course. And, if you multiply once again by -2, you have to do the same thing: change sides with respect to the origin and double the distance from the origin. You get 4, of course. If we multiply twice by -2, we have multiplied by 4. Multiplying by -1 is very easy. Each point is sent to the symmetrical point with respect to the origin, in other words we do a half-turn, a rotation by 180 degrees, if you like. When we multiply a number by itself, the result is always positive. For instance, if we multiply by -1, we make half a turn; so that if we do it one more time, well, we come back to the initial point! This is why -1 times -1 equals +1 simple enough. You see for instance that multiplication by -1 sends 2 to -2 and that if you multiply one more time by -1, you come back to 2. Obvious, isn't it ? Therefore, there is no number which, multiplied by itself, yields -1. Another way of saying this is that -1 has no square root. But, of course, we are underestimating the inventiveness of mathematicians! At the beginning of the nineteenth century, Robert Argand had a really great idea. He said to himself: "Since multiplying by minus one is a 180 degrees rotation, its square root is a rotation by one half of 180 degrees: 90 degrees. If I do two quarter turns one after another, I end up doing a half turn! The square of a quarter turn is a half turn, hence minus one." It's easy when you know how! Argand decided therefore that the square root of minus one is the point where 1 ends up after a 90 degree rotation. But of course this forces us to leave our horizontal straight line, since we just agreed to assign a number to a point that's not on the line! Because this construction is a bit strange, we call this point, the square root of -1, an imaginary number and label it i. But, once we have the courage to leave the line, everything else is easy. We can represent 2i, 3i, and so on... Each point in the plane represents a complex number and conversely, each complex number defines a point in the plane. Points in the plane become numbers in their own right! These numbers can be added, just like usual numbers. Look at the red point, which is the point 1+2i. Let's add 3+i, which is the blue point. Well, you add them just as schoolchildren do, giving 4+3i. Geometrically, this is just addition of vectors. You see that it's no problem to add complex numbers. Much more interestingly, these complex numbers can also be multiplied just like real numbers. Let’s see We know how to multiply a complex number by 2 for instance. Two times 1+i, gives 2+2i . Geometrically, multiplying by 2 is easy: it's just scaling up by a factor 2: if we double the red point, we get the green point! Multiplying by i is not difficult either since we know that i corresponds to a quarter turn. In order to multiply 3+i by i, we just have to rotate by a quarter turn. We get -1+3i. Not so complicated, these complex numbers! And finally, we can multiply any two complex numbers with no problem whatsoever. Let's try for instance to multiply 2+1.5i and -1+2.4 i. We proceed as usual, we first multiply by 2 and then by 1,5 i then we add the results. Therefore we get: "Two times etc.." Hence -2 + 4.8 i - 1.5 i + 3.6 i×i But, recall that i squared is -1, since we invented i for this purpose! This gives: -2 -3.6 ..etc, Let's clean up a bit. We find -2 -3.6 + 4.8 i - 1.5 i, that's -5.6 + 3.3 i. There you are, we know how to multiply complex numbers in other words, we can multiply points in a plane! Amazing! We thought that the plane was 2-dimensional since two numbers are necessary to locate a point and now, I'm telling you that one number is enough! Of course, we changed our numbers and are now dealing with complex numbers! Now's a good time to define two notions: the modulus and the argument of a complex number. The modulus of a complex number z is just the distance from the origin to the point that represents z in the plane. Let's use the ruler to determine the modulus of the red point which is 2+1.5 i Let's see, it measures 2.5. The modulus of 2+1.5 i is therefore 2.5. For the blue points, I get 2.6. And for the green point, which is the product of the two points, I get 6.5. What's the rule? The modulus of a product of two complex numbers is just the product of the moduli of the two numbers. The argument of a complex number is measured by the angle between the abscissa axis and the straight line joining the origin to the point. Here for instance, the argument of the red complex number is 36.8 degrees. The argument of the blue point is 112.6 degrees. And for the product, the green point, we get 149.4 degrees: this is the sum of the arguments of the two numbers... When we multiply two complex numbers, moduli are multiplied and arguments are added. Let's finish up our first encounter with complex numbers with stereographic projection. Consider a sphere tangent to the board at the origin. Using stereographic projection, each point on the board, that is, each complex number, corresponds to a point on the sphere. Only the sphere's north pole, I mean, the pole from which I'm projecting, has no complex number associated to it. We say that it corresponds to infinity. Therefore, mathematicians say that the sphere is a complex projective line. Why line? Because one needs only one number to describe its points! Why complex? Because this number is complex. Why projective? Because we added a point at infinity, using the projection. Aren't mathematicians strange when they try to tell us that the sphere is a straight line?

[6] Complex numbers...continued I'm going to show you some transformations. Transforming what? Well, if you don't mind, we'll transform my portrait. Let's begin with something simple: the transformation z goes to z/2. Each point on the photo corresponds to a complex number z that's divided by 2. We get another point, its image by the transformation, hence a new picture. You see, no surprise, I just shrank to half my size since each z has been divided by 2! This transformation is called a dilation. How about the multiplication by i ... Easy! We know that multiplying by i, is just a quarter turn. Note that the modulus doesn't change but the argument increases by 90 degrees. Indeed this is quite a sophisticated way of saying that we just rotated the picture! Well, a bit more complicated... Multiplication by 1+i. Look at the complex number 1+i , corresponding to the point with abscissa 1 and ordinate 1. Its argument is 45 degrees and its modulus is the square root of 2, using the Pythagorean Theorem. Hence, multiplication by 1+i amounts to first multiplying its modulus by the square root of 2, and then adding 45 degrees to the argument. In simple words, you have to combine a dilation and a rotation. This is called a similarity. More interesting! We are going to transform the points z into their squares z times z. Let's begin by placing the photo in a suitable place; flush against the coordinate axes. Then, I zoom a little bit since the squaring process will change the size of things and I need enough space to show you this. OK, now, we can transform the photo progressively. Notice that the argument of z squared is twice the argument of z, so that the right angle on the lower left of the photo is doubled under the transformation. It's turned into a 180 degrees angle. Let me place the photo somewhere else and let's look again at the same transformation z squared. You'll notice again the same argument doubling. Look for instance at my index finger. Before the transformation, its argument is about 45 degrees and after the transformation, it points upward, at 90 degrees. But you can also observe that moduli are squared. Now let's go to a new transformation, sending the point z to -1/z. Don't forget, with complex numbers, one can add, multiply, but also divide (not by zero, of course!) Doesn't this image remind you of the Sistine Chapel? Large complex numbers, with a large modulus, become small when one takes their inverses, and conversely. Here is a similar transformation. Look at the formula. The value of k changes slowly. Some parts are expanded, others are contracted, but if one looks closely, the shape is preserved, even though lengths are not. A circle remains circular, even though it might grow: my hand grew, my face became smaller but you can still recognize me! One more transformation, more involved. Well, this one's not really... a weight-loss program for me! But note once more that, even though I got bigger, the shape of small parts did not change: if you look for instance at a button on my shirt; it keeps a circular shape. One says that these transformations are con-formal or holo-morphic, rather complicated Latin and Greek words for saying that one preserves shapes! Indeed, with complex numbers, one can do quite a lot; one can even take the exponential, if you know what this means! But, even if you don't know, look at the treatment I have to suffer from the exponential! Has my head disappeared? No! If you looked through a microscope, near the origin, you could see my beard. Now that you know about complex numbers and have seen some transformations, I'll explain some of the objects I've been studying. Here, you see a number of points Some are blue, inside the unit disc, and some are yellow, outside. Let's perform the transformation z squared several times and look at the result. You can see that the blue points stay inside the disc while the yellow points escape from the disc, and even escape from the screen. One says that the blue disc is the filled-in Julia set of the transformation z squared. Points outside the Julia set escape to infinity when one repeats the transformation indefinitely. But, we can play the same game with other transformations; like for instance those of the form z squared plus c where c is a complex number, that we can choose at will. For each complex number c, we get a Julia set whose shape changes when c changes. You can see a few examples here. Here's the one I called "the rabbit"! In order to understand how these shapes change, I'll show you several things in parallel. On the left hand side, the red side, you can see a point that will start to move: this is the point c. On the right hand side, you see the corresponding Julia set: it's deforming as c slowly changes. But sometimes, for some values of c, the Julia set seems to disappear, and we can't see anything on the screen anymore, like now, for instance. The truth is that the Julia set blew up into an infinite number of pieces so small that you don't see anything on the screen. Benoit Mandelbrot, who popularized fractal sets, suggested the study of this set, drawn in red, that describes the values of c for which one can see the Julia set clearly on the screen in other words, those for which the Julia set does not blow up in multiple pieces. Of course, this red set is called the Mandelbrot set, and I spent quite a lot of time studying it. To finish, let's look closely very closely, at this Mandelbrot set, and zoom inside so you can appreciate how beautiful it is... Let's go, here it is! Admire... For once, I won't explain everything. Imagine the Mandelbrot set as a black island, surrounded by a tropical sea, and that you can see the bottom of it. Really, you're looking at truly microscopic details... If the Mandelbrot set were the size of a soccer field, we'd be looking at details of the size of a single atom, on the order of millionth of a millimeter! Maybe you're wondering why I got interested in this? First of all, because it's beautiful and because understanding these objects gave me much pleasure. For me this is reason enough to spend time on these questions. But also, because in these transformations that look so simple; one can find the essence of chaos, such a fundamental concept in modern science. Simple things generating rich structures! To study complicated phenomena through their simplest incarnation, this is often the role of the mathematician.
[7] Fibration Circles in space... arranged to create beautiful ornaments. In order to better understand the 3-dimensional sphere in 4-dimensional space, I'm going to show you how to fill space with circles and thus create what mathematicians call a "fibration". By the way, my name is Heinz Hopf and I'm one of the main contributors to the development of topology during the first half of the twentieth century. Look at this torus, filled with circles that appear to be linked. Let me explain this picture to you. Circles, spheres, and tori are among the simplest objects studied by topologists. A topologist tries to understand the connections between these objects. I worked in Berlin, Princeton and Zurich, and one still comes across my name often in contemporary mathematics: Poincaré-Hopf theorem, Hopf invariant, Hopf algebra, Hopf fibration. Let me paint my portrait for you. I published the discovery of "my" fibration in 1931 but, as always, I have to say that I relied upon many predecessors, like Clifford for instance, whom you see here, and who worked in England during the nineteenth century. Let's begin with some explanations on a blackboard, well a whiteboard this time! What do you see? A 2-dimensional plane? Well, yes and no! This is indeed a 2-dimensional plane but it's a plane with 2 complex dimensions, or in other words, a space with 4 real dimensions. Give it a try! Each point in this plane is determined by two coordinates; but each of these two coordinates is a complex number, which, remember, is itself defined by two real numbers... Each of the axes is a complex line, so each point on a given axis has one coordinate which is a complex number. For instance, here you see the point 2-i on the first axis. The same is true for the other axis, the y-axis. Here, we can see the point 1-2i on this axis. Now our whiteboard is magical, but not enough to be able to show us the two planes simultaneously! If we try to depict them in 3-dimensional space, they will intersect along a line but in 4-dimensional space, they intersect only at the origin — after all, they are axes! Now what do you see? A circle? Yes... and no! What you see, or rather, what you should imagine, is the set of points in 4-dimensional space that are a distance 1 from the origin. In other words, this is nothing other than the 3-sphere S³. Well, of course, you need to have a little imagination... Let's try to see at least how this sphere intersects the first axis. The 3-sphere intersects the first axis in the set of points on this axis which are at distance 1 from the origin. You see, the 3-sphere intersects the first axis in a circle. The same is true for the second axis which intersects the 3-sphere in a circle as well, the blue circle. Now, what is true for the horizontal line and the vertical line is equally true for all lines going through the origin. Here you can see the line with equation z2=-2z1 but we could do the same with any line z2 = a z1, for any complex number a. In this manner, the 3-sphere in 4-dimensional space is filled with circles; one for each complex line going through the origin in our plane of two complex dimensions. Careful though! In the picture, you get the impression that the red circles intersect each other but this is not the case in the 4-dimensional reality. Lines meet only at the origin, so their intersections with the unit sphere don't intersect at all, in fact. I discovered some very interesting things about this decomposition of the sphere into circles. and ever since, it is known as the Hopf fibration. Why fibration? Well, you should think of the fibers of fabric. We are going to look at all that using stereographic projection. Imagine that we project the 3-sphere from the north pole onto the tangent space at the south pole, which is our 3-dimensional space. Here is the projection of one of the circles which as we have seen, is the intersection of one complex line and the 3-sphere. But there are many such circles, one for each complex line going through the origin. For each complex number a, we can consider the line z2=az1 and its associated circle. Let's vary this number a or, what will amount to the same thing, let's rotate this line in order to see how the circle changes. Notice that sometimes the circle appears to be a straight line but this is simply because it passes through the north pole of our 3-sphere. Let's look at two of these circles simultaneously. In the lower left-hand corner, you see two moving complex points, one red, the other green. You can see the circles associated to the red and green points. Note that these two circles are linked together like two links of a chain: it is impossible to separate them without breaking them. Just for the fun of it, let's consider three circles... See how the three linked circles dance together. Now let's take many more complex lines, chosen randomly, and let's look at them all at once. The circles fill space, and no two of them intersect: this is an example of a fibration. Let's try to understand this better by returning to the board for a moment. Look, we have a Hopf circle for each line Each one of these lines has an equation of the form z2 = a z1 where a is a complex number, the slope of the line, and is indicated by the red point moving on the green line... Actually, the vertical axis does not have such an equation but in this case, we may say that a is infinite. Don't forget that a is a complex number. The green line is also a complex line, so it's a real plane, of course. Summing up: the complex lines that we are interested in are completely determined by a point on the green line and an additional point at infinity. But we already saw that if we add a point at infinity to the complex line, we get the usual 2-sphere. Once more, this is stereographic projection. So the complex lines that interest us are parameterized by points on the yellow sphere; the 2-dimensional sphere S². So we have a circle for each point on the 2-sphere. But a circle is a 1-dimensional sphere, isn't it? All these circles fill up the 3-sphere. Each point on the 3-sphere belongs to a single circle and therefore defines a point on the 2-sphere. In this way, we get a projection from the 3-sphere to the 2-sphere. Complicated, isn't it? Mathematicians say that above any point of the base S² there is a fiber which is a circle S¹ and that the total space of this fibration is the sphere S³. I am very proud of my fibration all the more so because it has become a fundamental object in topology!
[8] Fibration...continued. Let's come back to the 2-dimensional sphere and its parallels. Above each point of the 2-dimensional sphere, we should imagine a Hopf circle. Look at what's above one of the parallels of S², the equator for instance. Here's what's above another parallel which is moving southwards. Why does the torus seem to get thinner? Because above the south pole, there's of course only one circle. and above the north pole, you see a straight line, actually a circle, going through infinity. This is the red line! Well, let's spin all this around now. Rotations, yes, but rotations in 4-dimensional space of course. To be honest, I must admit that some of these pictures were already known long before me. The existence of four families of circles on the torus is usually attributed to the Marquis de Villarceau but one finds earlier clues, in a sculpture in the Strasbourg cathedral for instance. Take a torus of revolution: this is the surface described by a circle rotating around an axis in its plane. Slice the torus with a plane. Notice how I chose the plane. One says that it is bitangent to the torus, simply because it is tangent at two points. Now look carefully: the plane cuts the torus along two perfect circles. This is Villarceau's theorem: a plane which is bitangent to the torus cuts it along two circles. Of course, there's not just one bitangent plane. Here's another one, cutting the torus along two other Villarceau circles. And we can do the same for all other bitangent planes: we just need to rotate around the axis of symmetry. You see, through each point on a torus of revolution we can draw four circles, obtained by suitable slices. One of these circles is a parallel, another is a meridian, then a first Villarceau circle and a second one. And since one can do this at any point of the torus, we see that the torus is covered by four families of circles. Two circles of the same family do not intersect. A blue circle intersects a red circle in a single point. A yellow circle and a white circle intersect in two points: these are the Villarceau circles. Take a good look at the yellow circles: these are Hopf circles! Remember when we looked at what is above a parallel in the fibration? We saw a torus covered with linked circles, just like this torus covered with yellow circles. And what about the white circles? Well, they are the fibers of another Hopf fibration! ...the mirror image of the first one. To finish our stroll, we'll take a torus of revolution, with its four families of circles; imagine it in the 3-dimensional sphere, rotate the torus inside the 3-dimensional sphere, and finally project it stereographically onto 3-dimensional space. In this way, we obtain surfaces that are also covered by four families of circles: the so-called Dupin cyclides. Sometimes, when the torus passes through the projection pole; the surface becomes infinite... In this movement, the two faces can even be switched. The inner face of the torus is pink and the outer one is green. A simple rotation in the fourth dimension and …bingo! green turns into pink and pink into green. Isn't that magnificent?
[9] Coming soon! "Dimensions" Part Two! You'll see that dynamics is the science of movement.. ..that topology is the science of shapes.. ..that arithmetic is the science of numbers. You'll discover how topology can explain dynamics.. ..but you'll also learn how numbers can start moving.. ..or how numbers in movement can create incredible topologies. Yes, you'll see with your own eyes one of the most difficult problems in mathematics yet to be solved: the Riemann hypothesis. Yes, what you see here illustrates what mathematicians call "the horocyclic flow". But to understand all these things you will have to wait for the next DVD, or maybe the next several DVD's! There is so much to show in mathematics — see you soon!