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En este episodio. Senorita, ¿a quien busca . Por ejemplo, ¿quien es esta senora . ¿que le estara diciendo a raquel . Y esta senorita, ¿quien sera ella . ¿tiene informacion sobre Angel Castillo . Tambien vamos a aprender el vocabulario relacionado con los mapas de una ciudad y como seguir instrucciones. Raquel esta aqui en la esquina. Ella debe virar a la izquierda. Luego debe seguir derecho hasta la bocacalle. Luego debe virar a la derecha. Y tambien vamos a aprender Ciertas Cosas sobre la geografia de puerto rico. ¿que hace ud. Aqui . Estoy tomando una foto. ¿de la tumba de mis padres . Captioning of this program is made possible by the Annenberg Cpb project and the geraldine r. Dodge foundation. En el episodio previo, raquel se despidio de arturo iglesias, el medio hermano de Angel Castillo. Raquel y arturo sienten un afecto muy especial el uno para el otro. Y la despedida fue triste, especialmente para arturo. Raquel hace otro largo viaje para llegar hasta aqui, san juan capital de puerto rico. San juan queda en el norte de puerto rico en la costa del atlantico de la isla. Otras ciudades importantes son caguas en el centro de la isla ponce en el sur en la costa del caribe y mayaguez en el oeste. Pero san juan, por ser la capital es la ciudad principal de la isla. En realidad, cuando se habla de san juan se habla de un san juan moderno, con edificios altos y ciudades vecinas como hato rey rio piedras, donde esta la universidad y santurce y de la famosa zona turistica, la playa del condado y tambien dEl Viejo San Juan, el san juan historico. Aqui en El Viejo San Juan hay iglesias. Casas. Murallas y edificios, todos de la epoca colonial. Y es aqui, en El Viejo San Juan donde raquel comienza a buscar a Angel Castillo. Ique es esto i¿como me estan haciendo esto que todas las calles esten en estas condiciones . imire esto ¿que diablos estan haciendo aqui que todas las calles estan bloqueadas . No me importa caminar de aqui si no esta lejos la calle sol. ¿que numero busca ud. . El cuatro. Mire, ¿ve la esquina . Si. Tome a la izquierda. A la izquierda. En el proximo bloque, vire a la derecha. A la derecha. Camine derecho hasta que encuentre unas escaleras a la izquierda. A la izquierda. Baje Las Escaleras y cuando encuentre la calle sol. ¿cual es el numero que busca . El cuatro de la calle sol. Entonces, creo que esta a mano derecha. Cuando encuentre la calle sol si se pierde, pregunte. Todo el mundo conoce esa calle. ¿cuanto le debo . El metro marco cinco dolares. Mm. Aqui tiene. Muchas gracias. No hay que hablar. Que le vaya bien. Gracias. Raquel esta aqui en la esquina. Ella debe virar a la izquierda. Luego debe seguir derecho hasta la bocacalle. Luego debe virar a la derecha. Debe seguir derecho hasta Las Escaleras. Debe virar a la izquierda y bajar Las Escaleras. Al final de Las Escaleras, debe virar a la derecha. Disculpe. Estoy buscando la calle del sol, Numero Cuatro. La calle del sol, es esa que esta en el frente. El Numero Cuatro es a la derecha. ¿a la derecha . Exacto. Gracias. De nada. timbre suena senorita, ¿a quien busca . Buenos dias, senora. Busco al senor Angel Castillo. ¿no sabe ud. , senorita . El senor castillo murio. ¿cuando murio . Hace poco. Es una pena, tan Buenos Vecinos que eran. Pero el pobre. ¿angel . Si, Angel Castillo. Nunca se repuso de la muerte de su esposa. ¿entonces era casado . Si, su senora era una mujer muy linda. Era escritora. Pero murio ya hace varios anos. Los dos estan enterrados en el antiguo cementerio de san juan. ¿en El Cementerio . Si. ¿y podria decirme como llegar alli . Por supuesto. Siga por esta calle. Entonces, vire a la izquierda. Luego va a encontrar una bocacalle y vire a la derecha, y alli esta el morro. Al lado esta El Cementerio. ¿sigo por esta calle, luego a la izquierda encuentro una bocacalle, alli a la derecha alli esta El Morro Y Al lado El Cementerio . Asi es. Muchas gracias, senora. Por favor, senorita, ¿quien es ud. . Una amiga de la familia. Buenos dias. Buenos dias. Perdone. Busco la tumba de Angel Castillo. Siga ud. Derecho, hacia la capilla alli doble hacia la derecha. Tres lineas mas, la encontrare. Gracias. Perdone. ¿que hace ud. Aqui . Estoy tomando una foto. ¿de la tumba de mis padres . ¿de sus padres . Si. De mis padres. Perdoneme pero. Tengo que sentarme. ¿se siente mal . ¿por que no nos vamos a la sombra . Entonces, raquel y la senorita comienzan a charlar. Raquel le cuenta la historia de Don Fernando Y Rosario y de su busqueda en espana y la argentina. La senorita se llama Angela Castillo y es la hija de angel. Angela no sabe nada de la historia que raquel le acaba de contar. Ino lo puedo creer itengo un abuelo que vive en mexico asi es, ud. Tiene un abuelo que vive en mexico. ¿y mi abuelo creia que rosario habia muerto . Exactamente. Y rosario tambien creia que fernando habia muerto. Ique historia tengo que llamar a mi familia. Y ud. Debe venir conmigo. Mi familia querra hacerle muchas preguntas. ¿vamos en Carro A Su Casa . Por favor, ¿nos tuteamos . Bueno. ¿vamos en Carro A Tu Casa . Raquel se sorprende al saber que van en carro porque las calles cerca de la calle del sol estan bloqueadas pero angela conoce otra ruta y no tardan mucho en llegar. Imilagro un lugar donde estacionar. Ique suerte Mi Apartamento esta a la vuelta. Cruzamos la plaza y luego vamos a la derecha. La vecina me dijo que tu mama era escritora. Pues si, mama escribia cuentos para ninos. Entra. Gracias. Itengo una sed increible voy a traer limonada para las dos. Iperfecto ah. Tienes un apartamento muy lindo. Gracias. Pero voy a mudarme pronto. ¿te mudas . ¿de aqui . ¿por que . No aguanto este lugar sin mis padres. Y tu, ¿como puedes viajar tanto . ¿no tienes problemas con la oficina . ¿yo . No. Este es un caso especial. Ademas yo llamo a mi oficina de vez en cuando. Voy a ver si estan mis tios. De acuerdo. Mmm. Ique magnifica vista si. Desde aqui tenemos una buena vista de la bahia. Iay ique maravilla sabes, raquel aquellos cuadros fueron pintados por mi padre. Ique lindos ¿puedo tomarles una foto . Si quieres. No estan. Todo esto me parece increible. Es la pura verdad. Quiero que le repitas toda la historia a mi familia. Claro. Es importante que lo sepan todo. Seguire llamando a mis tios. Entonces, mientras angela sigue llamando a sus tios por telefono raquel visita La Casa Blanca. Alli encuentra a un amigo de la Familia Castillo y el le muestra la casa. La casa blanca fue construida en mil quinientos veinte y uno para el famoso explorador espanol Ponce De Leon. Pero Ponce De Leon nunca vivio en esa casa. Murio en cuba en mil quinientos veinte y tres. Sus descendientes ocuparon la casa por doscientos cincuenta anos. En el siglo diez y ocho el gobierno espanol la declaro residencia militar. Y luego, en mil ochocientos noventa y ocho paso a manos del gobierno norteamericano. Ahora es un museo, donde se puede ver como era la vida diaria en la epoca colonial. La casa tambien tiene un balcon con una vista de todo El Viejo San Juan incluyendo una vista de la calle del sol. Alli a la Puerta De San Juan Donde Estamos ahorita. Ihola, turistas ihola, angela aqui estoy de guia en La Casa Blanca. Le encanta a tu amiga. No le creas lo que te diga, raquel. ¿conseguiste hablar con tu familia . Si. Con mi tia olga. Estan en camino. Ahora vuelvo. No, no. No te apures. Tardaran una hora en llegar. Como los tios de angela tardan en llegar angela decide llevar a raquel a otras partes dEl Viejo San Juan. Angela este es el parque de las palomas. Raquel ihay palomas por todas partes esta es la capilla del cristo. Tengo unas pinturas que mi padre pinto aqui. Vamos. No puedo creer que quieras vender este apartamento. Realmente es bonito. Ven. Sientate. Van a llegar pronto. Voy a llamar a mi novio, jorge. Esta ahora en nueva york. ¿en nueva york . Si. Jorge va a Los Estados Unidos con frecuencia. Perdona. Si, si. Aqui en san juan, comence la busqueda de Angel Castillo. Pero. ¿comence en el moderno san juan . ¿o en El Viejo San Juan . Comence la busqueda en El Viejo San Juan. Varias de las calles estaban bloqueadas y el taxi no podia pasar hacia la calle sol. Entonces decidi seguir caminando. ¿quien me dio Las Instrucciones para llegar a la calle sol, Numero Cuatro . Mire, ¿ve la esquina . Si. Tome a la izquierda. A la izquierda. En el proximo bloque, vire a la derecha. A la derecha. Raquel el taxista me dio Las Instrucciones. Bueno. Por fin encontre la casa. Pero, ¿que paso . La vecina me dio unos datos muy importantes primero, que angel estaba casado y segundo, que angel murio este ano. Los dos estan enterrados en el antiguo cementerio de san juan. ¿en El Cementerio . Por fin llegue al cementerio. Encontre la tumba de Angel Castillo y de su esposa. Mientras yo tomaba una fotografia ¿quien llego al cementerio . Perdone. ¿que hace ud. Aqui . Estoy tomando una foto. ¿de la tumba de mis padres . Raquel Angela Castillo llego al cementerio. En casa, angela trato de hablar con sus familiares. ¿hablo con ellos o no . No estan. No. No los consiguio. Entonces yo sali para dar un paseo. Primero, fui a La Casa Blanca, un museo. Luego, fui con angela al parque de las palomas y tambien vi la capilla del cristo. Esta es la capilla del cristo. La visita a La Casa Blanca fue interesante. Era la casa de Ponce De Leon, el explorador. Pero Ponce De Leon murio en cuba. Nunca vivio en esa casa. Finalmente, volvimos al apartamento. Angela esta hablando por telefono. ¿con quien esta hablando . Bueno. Ahora es necesario que yo hable con la familia de angela. ¿como van a reaccionar ellos . ¿que van a decir . Angela no sabia nada de la historia de rosario y don fernando. Pero. ¿sabe algo la familia . Jorge no lo puede creer. ¿a que hora llega tu familia . Deben estar por llegar. ¿quien es . Ah, es mi hermano, roberto. ¿su hermano . Captioned by the Caption Center Wgbh Educational Foundation annenberg media for more information on the College Television course for information about this and other annenberg media programs call 1800learner and visit us at www. Learner. Org. Many things in the universe behave in a synchronized way whether manmade or natural working together in harmony, moving simultaneously, pulsing with a regular rhythm, coordinated in time and space. Moving in sync, we would say. We see synchronization as an emergence of spontaneous order in systems that most naturally should be disorganized. And when it emerges, theres a beauty and a mystery to it, qualities that often can be understood through the power of mathematics. How does a Symphony Orchestra play in sync and harmony . The answer is obvious the group consists of humans who have the capacity to read music, listen to each other, and follow their leader. But what about a group with no leader or sheet music . Like a jazz jam session . Itstill perform in sync, each performer working off the cues of the others, synchronized via spoken or unspoken communication. What about schools of fish or flocks of birds . How do they know when to turn left, turn right, move higher or lower, perfectly in sync, each adjusting its movements instinctively, somehow highly sensitive to what its neighbors are doing . While the orchestra and marching band work in a premeditated, planned, even calculated order, the flock of birds and school of fish, they move in unison by a phenomenon called spontaneous biological order. In their collective movement, we see a system of individuals who somehow have managed to synchronize their changes in motion. And through mathematics, we can make sense of this Group Dynamic using the language of calculus. To put it simply, calculus allows us to make mathematical sense of change in moving systems, whether gradual or constant. For instance, a cars speedometer may say 55 miles per hour, but thats just a rough approximation of its exact speed at that moment. How can we describe exactly how fast the car is moving at each instant it drives along . For this answer, we must go back to the 17th century, when the english scientist robert hooke challenged his rival isaac newton with that kind of question. He asked how exactly do the planets pull on each other, and does the law of pulling explain the orbits that we see . To answer it, newton realized that he had to describe and measure the movement of the planets instant by instant. And so, in 1666, he independently creates a new branch of mathematics, calculus. Newton needed a way to notate the incremental changes of where a planet was on its orbit and then to be able to add up all of those incremental changes to actually construct the orbit. He also needed a way to figure out how fast it was moving and where it would move next. He saw unity; gravity didnt end at the atmosphere. Newton was the first to show that the motion of objects on earth and planetary motion are governed by the same set of natural laws. His famous equation force equals mass times acceleration, f ma, neatly sums that up. This same math he used to study the movement of inanimate objects in the 17th century today assists us in calculating the behavior of animate objects. Thanks to newton, calculus provides a mathematical language that allows us to measure a variable. For instance, for a bird, the angle of flight, or for a car, its speed. But birds and drivers are their own entities with reactions and influence factors, and therefore represent moving objects with many variables. Its perhaps a bit easier to learn about calculus by examining the solar system, because with planets theres a consistent measurable force at play. In order to study the rate of change of an object in motion, like a planet, we look at the tiny changes in the position of the planet in the sky. These tiny changes are then called infinitesimals. They are a theoretical construct and are meant to be the smallest possible increment by which a quantity can change. This idea leads us on to derivatives, another important concept in differential calculus. Derivatives allow us to describe how fast something is changing at any particular instant. In the case of a planets orbit around the sun, lets say that its position at any moment in time is described by a variable. Well call it p for position. Since this position changes over time, we write this as p t , or p of t thats how mathematicians say it which simply means position at some time, t. Now, in some tiny interval of time, described by dt, the position will change by some tiny amount, which we describe as dp. If we compare the small change in position to the small change in time, we get a ratio, dp dt, and this is like an instantaneous and infinitesimal calculation of an average velocity how far did i travel in how much time . So this ratio is the derivative, and it tells us how fast something is changing at any particular moment. In this case, it tells us how fast the planets position is changing with time, which we commonly refer to as the planets velocity. So if we use this ratio dp dt, which represents the planets velocity, to write a calculation of its velocity in relation to the forces of the sun, we create a differential equation. We call it a differential equation because its an equation relating very small differences or differentials. This kind of equation describes how the planets motion changes dependent upon its relative distance to the sun and the other planets. Differential equations can be used to understand more than just the movements of the planets. They can be used to study other systems where physics describes the pushing and pulling of objects on each other. Often the scale and complexity of these systems are such that it is only through mathematics that we can simplify and understand them. This simplification will take a big system and turn it into a system of coupled differential equations. These can then be investigated on the computer. Although calculus was developed out of newtons desire to explain the movements of celestial bodies, the same mathematics can be applied to whole realms of science, from Solar Systems to the beating of our hearts. My names charles peskin, i am a professor of mathematics at the Courant Institute of mathematical sciences, new york university. What i do is math and computing applied to biology and medicine. And my kind of main project is the heart, a computer simulation of the heart. The question is how can we understand the basic laws that govern the heart well enough that we can make a model heart in a computer that works the way the real heart does . So the equations which i use to describe the heart are differential equations, so calculus is absolutely fundamental to what i do. The heart has its own pacemaker, its called the sinoatrial node. Its a clump of cells which send out waves that synchronize the heart. And whats amazing about these cells is if you grow them in tissue culture separately, they beat on their own, but theyre not synchronized. And then when they grow and come in contact, they synchronize with each other. So the question is basically how does synchronization work, and whats involved in synchronization . Its an amazing fact that i dont know exactly how to explain, but its mysterious and interesting, that when you get spontaneous oscillators that each of which has its own rhythm, when theyre coupled together in some way, when they influence each other, even if the influence is very weak, they have a tendency to synchronize. Im very fortunate to have the opportunity to work with glenn fishman. He studies electrical conduction in the heart. My name is glenn fishman, im director of the division of cardiology and head of the Cardiovascular Biology Program here at nyu school of medicine. Our lab group is interested in understanding the basis for cardiac arrhythmias. Weve taken this out of the mouse and hooked it up through the aorta to profuse it, to keep it happy. We can look at that both in normal conditions as well as in some of the genetically engineered mice that weve made without gap junctions or with other channel abnormalities and try to understand why they get arrhythmias, which is really the main question thats driving us here. If we understand the biology of the pacemaker cells, its our hope that we can regenerate portions of the conduction system by reimplanting cells that take on this function from those that have degenerated in the patient. More broadly, in terms of the whole hearts electrical system, if we can understand the basis for many forms of lethal cardiac arrhythmias, we can go in and treat those sorts of diseases. Cardiovascular disease is the leading cause of mortality in the united states, and sudden cardiac death from arrhythmias is the leading cause of death within the cardiovascular category. Its clearly relevant just in terms of the Public Health burden that we see from cardiovascular diseases. People want to just be able to say some words which explain what happens, but actually what you need is equations and mathematical models and computer simulations, because if you didnt know the basic rules that govern it, you could not have made the model. [ sound of heart beating ] it takes 10,000 cells in a tiny corner of the heart, all working in sync to keep us alive, to keep a steady rhythm inside us. Each cell is a bioelectrical oscillator, similar in concept to the mechanical oscillator of this clock, meaning that they each obey a basic rhythm. And that rhythm can be understood using the language of calculus. For the clock, the state is the position of the mechanism, and for the heart cell, its how close it is to firing an electrical signal. However, mathematically, these different mechanisms are the same. So any single oscillator has a simple description, but now were interested in a much more complicated problem. What happens if theres a group of oscillators with a means of communicating with one another but with no Single Authority dictating an overall plan, be they oscillators or birds, heart cells or fish. Can differential equations help us understand those kinds of phenomena . So to get to the answer, were going to talk to steve strogatz, a professor of applied mathematics at cornell university. Steve, how do we use math to get from heart cells to pendulums . Theyre both oscillators. They both move in cycles. You know, theres an electrical cycle in the heart cell where its voltage goes up and back down, and up and back down. So thats a rhythm, thats a cycle. And, of course, a pendulum is a cycle. Right, back and forth. In that it swings back and forth, so theyre really not that different. I mean, it happens one is an inanimate thing, one is living; that doesnt really matter to a mathematician at the abstract level. What we love in math is that theres a unity, that you can see the connection between pendulums and heart cells if you maybe have the same equation that can describe both, but just with different interpretations of what that just one abstract framework, and then you fill in the phenomenon for the variables. Right. Thats what we like to do. And in the case of pendulums, which we understand much better than heart cells, it gives us a way in to understanding these mysterious phenomena of life by thinking about mechanical things that weve understood for several hundred years. So one connection that we can make here is when we think about an enormous collection of cells in the heart, this is you could think of it as a kind of population of oscillators. That is, theres maybe 10,000 of these cells in the case of the pacemaker of the heart. And were going to describe each of them by a differential equation that tells how it changes its voltage from time to time. But whats difficult about this is we need to understand their behavior as a group. Its not enough to look at each cell in isolation. Because each cell in isolations actually pretty well understood, right . That would be easy. Understanding one oscillator, one rhythmic thing, is no problem. And, as i say, the challenge is to understand the cooperative or collective behavior of hundreds or thousands of them. This really was an outstanding challenge, even as late as 1960. In this case, art winfree was then a College Senior at cornell, interested in biology. He knew that he was interested in it, but he was in the Engineering Physics program. So anyway, winfree thought about this question of synchronization of it could be heart cells, it could be fireflies flashing, it could be crickets chirping, cells in the brain. He abstracted all of them as these differential equations, math that describes rhythmic motion, and put in the computer thousands of them together. So in his math he allowed for some of these oscillators to be inherently faster or slower than others. So that, you see, is a challenge for synchronization, because how are the slow guys going to keep up with the fast guys . And the answer is, because they Pay Attention to each other. Now, what does that mean for a cell, which cant think . It means that it can feel electricity, in the case of the heart. Cells send each other electrical currents that can cause one to fire faster than it would have otherwise or that can retard it. So by this chemical and electrical communication, or the math that corresponds to that sort of interaction, winfree was able to make this population behave as a cohesive unit. Sometimes. Heres a metaphor. Imagine that you have runners on a track. Okay. Its a track like youd see at a football stadium. So maybe a track like this one. Now, in any system where youve got runners, there are going to be fast ones and slow ones, and theyre going to be analogous to our inherently faster or slower oscillators. And were going to colorcode them so you can see whos inherently fast or slow. First, if theres no interaction, theyre ignoring each other. And you can see that the fast ones are running away. Theyre running away from the slow ones. Okay, but now suppose that its a running club rather than people who dont know each other. And maybe, you know, like you i know youre a pretty good runner, and im not. But maybe if you and i were running, i would want to keep up with you. And id want to be a social guy and i wouldnt want to outpace my buddy, sure. Right. So thats coupling, okay, that youll slow yourself down as you feel me on your shoulder, and ill be huffing and puffing to keep up. Now, in winfrees computer simulation, he imagines conceptually turning up the knob so that now maybe they care about each other more, or theyre listening. Like maybe theyre shouting to each other, hey, speed up slow down so as that interaction strength builds up, at first nothing happens; theyre still desynchronized. Thats a little bit surprising. You might think that with more interaction, theyd get a little synchronized, but they dont. Nothing happens until a critical Phase Transition point is reached, a kind of tipping so even though theyre talking, the fast guys are still going yeah, that was a little bit of a surprise. I mean, normally in this world you think that if you change one thing, it produces a response in something else. So here, if you would increase the strength of how much they care, they should start clumping more, but they dont. And thats what we mean by Phase Transitions. Like you can cool water, and its still water. And you can cool it more, and its still water. But then when you hit the magic point, in that case, freezing point, it changes qualitatively. It becomes ice. And thats what happens here. Now, heres the second scenario. Now weve crossed this tipping point, the Phase Transition, and you see, theres a little clump. The middling oscillators, the guys that are not too fast and not too slow and there are a lot of them start to lock together, run as a group in lockstep. And then finally, if we make the coupling Even Stronger so im turning up this conceptual knob even more now the whole population starts to run in sync, but theres some distribution which we can draw as a curve. And what is this curve showing . Its like in the case that were talking about now, how fast are you, as a runner, would be graphed on this axis. And then how many people are that fast would be graphed on this axis. And the reason its bellshaped is that most people are in the middle range of speeds. And there are some very fast, but not so many, and some very slow, but not so many. So you have this characteristic bell shape. So you really need probability statistical tools to understand life phenomena, right . Yes. And thats what makes them so hard mathematically, that you have to combine many things. And we see this personified by art winfree himself, that he was able to use Probability Theory to talk about the distribution of the speeds. Fantastic work. The most beautiful thing about this, or one of the great things, is that as much as this is about heart oscillators or runners, its actually about a multitude of things, because mathematics is this language that can describe everything by putting different phenomena into the variables. [ playing Classical Music ] humans, animals, and even our own heart pacemaker cells are capable of, and have a natural tendency toward, synchronization. But does synchronization exist on a nonbiological level . Could the impulse toward spontaneous order be more fundamental than what seems apparent in the biological world . Christiaan huygens, the dutch mathematician, physicist, and astronomer who invented the pendulum clock in 1656 made a startling discovery about his clocks while ill and bedridden. Noticing two adjacent pendulums swinging in perfect opposition, he wondered what caused this surprising form of synchronization. Was there some Mysterious Force locking them in rhythm . Huygens discovered that the clocks, if taken out of synch, created a disturbance, making the surface on which they rested tremble. Then something remarkable happened. The pendulums began sync up in their rhythm, swinging precisely opposite each other like a pair of clapping hands. Huygens assumed that the proximity of the clocks to each other and the air disturbance they created was responsibible r synchronizing the pendulums. To test his theory, he placed the two clocks on a plank, and the plank lay atop two chairs positioned back to back. He disrupted the pendulums sync as before, and immediately the chairs and the plank began to shake. The physical disturbance operating on the chairs continued for another 30 minutes, until the clocks restored themselves and the chairs and plank stabilized. Huygens discovered that the rocking disturbance caused by the pendulums swinging out of sync eventually brought them back into sync. The explanation for this involves the equal and opposite forces in play. When the clocks were in sync, the force they exerted onto the chairs and plank canceled out any physical disturbance that might have happened. Once that force was disrupted, the trembling started. So the movement of the chairs and the planks stabilized the pendulums, and then that in turn stabilized the chairs. Is that right . These pendulums swing as they will, and as they if theyre swinging in some sort of strange way, not synchronized, what happens is that they end up putting peculiar forces on the plank that makes the plank jiggle, and the chairs start clattering on the floor. Now, that then back, in turn, puts forces on the pendula that affects their motion, and the whole thing eventually settles down to a state that is, theres negative feedback onto the pendula, thats really whats happening, until the pendula get like this, antiphase, as we saw, okay . And when the pendula are like that, theyre not putting any net force on the plank, because when im pushing this way, youre pushing that way, and it cancels. So the plank gets still, the chairs get quiet, and we have synchrony. Voila, yes. So negative feedback through the pendula onto the planks and the chairs is thats what stabilizes the system. This is making me think of this very recent sort of, well, near disaster that that we saw in london, right, with the millennium bridge. Is that right . All right, watch. Londons millennium bridge, 325 meters long, links the city of london at st. Pauls cathedral to the tate modern gallery across the thames. It opened to the public on june 10, 2000. Designed by architect lord Norman Foster with sculptor sir anthony caro and the Engineering Firm arup, it was described as a blade of light that would cross the thames and create an absolute statement of our capabilities at the beginning of the 21st century. Approximately 80,000 people crossed the bridge in the opening days, many more than anticipated. Early on, vibrations of the bridge were being detected by the visitors. The engineers planned for some natural movement, of course; the structure was designed for it. But the bridge began to wobble and sway, enough to concern public officials, so that they eventually closed it. Thousands of londoners showed up to walk across this beautiful new footbridge, the millennium bridge, and you can think of those people as analogous to the pendulums. Except can i say one thing thats different . Absolutely. One thing thats going to be different is that these pendulums that is, these people arent going to stabilize this bridge, this plank; in fact, theyre going to destabilize it. I think the thing thats important in keeping in mind here thats different is that people dont like to walk on something thats wobbling. Pendulums dont think. Right. But people do, and people feel off balance and feel uncomfortable. Theyre actually reacting to each other and to the bridge. What was peculiar here is that as the bridge started to wobble, people did what they would naturally do to keep their balance, which is they separated their feet a little wider, they start walking like a novice ice skater, right, with their legs out, and they sort of have this penguin motion, which helps them feel steadier. But this is the part that no one would have expected by doing that, they started to pump energy into the bridge. They started to make the bridge move worse, which caused more people to adopt this weird gait, which pumped more energy into the bridge and made it worse. So you had this runaway feedback effect, which it turns out occurred through a Phase Transition very similar to what we talked about in the case of winfree and his biological oscillators. Except that here the Phase Transition had to do with the number of people on the bridge. After the problem happened, arup, the people that built the bridge, the Engineering Firm, put its own employees onto the bridge to try to diagnose what the problem was. First they put 50 of them and told them to walk around in a circle, and the bridge didnt move. And then they put 60, walking around, the bridge is still motionless. Somewhere between 150 and 160 people, the bridge started to move, the people started to walk like that, and so there was another, almost literally, tipping point, in this case. So this clearly looks like synchronization. So im assuming it is, and how do we get there mathematically . Well, it definitely is synchronization. We can see in that footage that you see whole sections of the crowd rocking from side to side in unison. So absolutely its synchronization among the people. Its also synchronization between the people and the vibrations of the bridge. Thats what makes this phenomenon possible, that the people are wobbling in order to stay comfortable in the way that they walk on this wobbling bridge. So we have two kinds of synchronization happening at the same time, and neither was anticipated. I would give the Engineering Firm, arup, a lot of credit because they figured out what was happening, to a large extent. Its not totally understood. Very mathematically sophisticated, in fact. One of the things they discovered through these studies where they put their employees on the bridge and caused them to excite the bridge in a controlled way was they found this very simple, shockingly simple formula, f k x v. Now, f just means the sideways force that a typical person, or the whole crowd, puts on the bridge, because thats whats driving the bridge. K is just a mathematical constant that relates these two different things, force and velocity. But what its saying is the more the bridge moves, the more force the people end up putting on the bridge to stabilize themselves, which ends up making the bridge move more. So that f k x v is the heart of the positive feedback loop that caused the bridge to start moving. So now weve been able to turn the interactions of the people and the bridge into mathematics, and this has now allowed the engineers to find a way to fix it, is that right . Right. Now, when a bridge is wobbling, engineers know there are really two things you can do to stop it. You can make it stiffer in some way by reinforcing it, putting trusses on it. They thought about it and decided that wasnt the right approach, for various reasons, but they were able to use the second standard strategy for curing a vibration, which is to make the bridge more heavily damped. You can put the equivalent of shock absorbers underneath the bridge. So they put Something Like 70 or 80 of these viscous dampers underneath the bridge; very unobtrusive so the bridge is still beautiful you dont notice and that solved the problem. And its just a terrific example of how rational thinking coming from math combined with engineering insight can cure what was this very peculiar, strange phenomenon. Math to the rescue. Yes. Its an uplifting subject, in a way, because it gives us a sense that there is theres the possibility for cooperation in the natural world, and its really built into existence. It comes as a relief sometimes when you think of, you know, all the disorder and disharmony that we see around us to think that there is at least this important side of nature where everything becomes harmonious. Terrific. Well, thanks so much, steve. Its been fun. Yeah, thanks. And so our world is a symphony of movement. When the power and beauty of the abstract language of mathematics of calculus and differential equations combine with the insights of the physical sciences, we begin to understand the how and why of that symphony, from the movement of the planets to the ticking of a clock to the flight of swifts to the beating of our own hearts. Captions by lns captioning portland, oregon www. Lnscaptioning. Com

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