Thank you very much, Jane. Thank you all for coming and people watching online. So we're gonna have fun today. Well, I'm going to have fun for sure. This year I want to uh talk about what I call the mathematics of surprise. What I will do is take a simple observation, simple phenomena, and see trying to understand it both on physical ground but also using mathematics and then see how that knowledge expand to other areas. And here's one of my favorite subjects, which is cracking the whip. But before we do that, I want to talk about something called the scientific methods, mostly organized by people who don't do science, as far as I can tell. And they are different versions, but they mostly look like that. You have an observation, a question, then you research a topic area, you formulate an hypothesis, you test it, you analyze the data, you report the conclusion and so on. And it feels a little boring, right? There's no like you wreak a moment, no excitement. And frankly, in my entire life, all the science I've ever done, and the people around me, nobody has ever followed this kind of wrong, right? So let me tell you about the other scientific methods. And usually it starts with observing something bizarre. You are in a field, and something is not quite what you expect. It doesn't match your understanding of that area. It's something extra, and you want to try to understand. So what do you do? Well, you gather useful and useless facts. Well, you don't know if they might be useful, you're just early days, right? So you go around and read books and watch TV, whatever. You do whatever you can to try to get. And then you turn to what I call back of the envelope computation. You say, okay, let's do some estimate to see if I get the right mechanism, maybe some order of magnitude to see how things work. And usually it's not enough, but that gives you an idea you're on the right track. And then you build a minimal mathematical model, what we call a toy model in the trade. Something that has all the components that is as simple as possible but not simpler, as Einstein would say. So it still captures something and you can learn from that. So here is a job both of modeling, and we talk about modeling, but also your job as a mathematician. No, you have a model, you have actual equation or something to that effect. You need to study the model, find solutions, see how the solution behave, maybe do numerical solutions on the paper, and so on, and you go around trying to get extract information of this simple equation. And after that, you go for a reality check. Does that match what you see? Does that match anything? And maybe it does, and then the big question, so what? What have we done? Right? And that you mostly don't by then, and you can write a paper, something like that. But of course, nothing is linear. And when you do your reality check, typically you start gathering other facts. Maybe you do and do little experiments or you try to find data, which forces you to maybe revise your model. Maybe you're missing an important mechanism in the model and so on. And after you think about it, you think, well, maybe there are other things that are like that and maybe a little different that follows the same pattern. And that forces you also to maybe write other models for other circumstances. And as soon as you gather facts, of course, you study the model, you have solutions, you have to compare them and so on, and it gets very messy. And every day you wake up and you're in one of these boxes and you jump from one to the other one. So today we start with something bizarre, right? And the something bizarre is the whip cracking. And it was bizarre to me as when I first thought about it. So I spent a long time in the southwest in Arizona, but that's not where I first uh realized that there was something interesting with the whip. It was in a meeting uh in Budapest in Hungary, and you know, uh there is usually a conference dinner in this meeting. It's a rather boring affair, big group taking a long time, drinking too much. But on this occasion, there was a folkloric band from Hungary coming and doing a little show with whip cracking, and I was there ignoring my fellows and looking and said, hmm, that's bizarre. Wonder what is this? You know, what is that so distinctive sound, a whip cracking? I knew a little bit of it before, but I got interested because at the time I was interested in dynamics of curve, and it didn't quite fit my understanding of that. So I started looking into that. But before we start, and to avoid any confusion, despite the movie Zoros and Indiana Jones, I want to let you know that the only, the one and only purpose of long whips, the one I'm going to uh talk about, is to make that sound, that crack. It's never been a weapon, it's never used for the purpose kinky otherwise. These whips, the one I'm going to show you, are only there to uh to create that whip. So I went home and it's time to gather facts. Okay, uh, or at least to try things. So the first thing I got, I got this uh big boy here, eight-footer. You know, it has a nice structure, we'll talk about that. And I started practicing, and I bought the bull whip book. Uh it was a time where you could learn skills from books. That was before YouTube, before even Google and stuff like that. And you learn in the books, very interesting, that there are three uh typical motions when you do that. There is three phases the setup, the turn, and the follow-through. So if you give uh a whip like this to somebody and you ask them to crack, they're probably gonna try to do something like that. So this one is too big to do here. So here I have a smaller one that I just got for for this purpose here tonight. It's only a four-footer. And so if you're given a whip, the first thing you'll try is what is called a snap, which is something like that. Yeah, you have to put a lot of force and you get a little crack, nothing impressive. Okay. And it turns out that yes, it works, you can do that. Um, but it's not the real way you really want to crack a whip. The way you want to crack a whip is to make a loop and let the loop propagate along. So I'm gonna show you have videos just in case it doesn't work here, but the the setup is very simple. You raise the whip up to a certain height and you turn it to make a loop that goes at the end. Right? And so if you do it like you see, right away, you create a crack with very little effort. And that gets very interesting, right? Because you send a loop and that loop accelerates and then crack at the end. Very distinctive, and so distinctive that it's not a normal sound, and that's why we use it, because animals usually never hear that kind of sound and they react to it, and that's what controls horses and cattle. And horses being the one method of transport to the 19th century, there was a huge, huge business associated with whip and cattle and ranching and all that. So, just in case it didn't work, I did my own uh experiment to show you. Uh, as you can see, I'm not only a scientist, I also play one on TV. So that's the whip, and here is a slow motion of the same action, and you can almost you can see the loop going through, right? It's high speed, and great, it works. Here is a uh younger version of myself in Arizona, and here is another way, it is this with this one. I'm not gonna do it here. I could almost maybe do it, but maybe I'll get in trouble. Uh and the idea is that you turn the whip around, so you create tension, that's a setup, and then you invert the motion to create that loop, and that loop zips to the end, and by the time it ends, it unfolds and cracks. So here is how it goes. You can see that the neighboring dog doesn't like it. And you can see here, you can see the loop forming there and going around. So you find in popular science book that the crack, that special cracks, occurs when the tip reaches the speed of sound. At the type where people were still looking for books and stuff like that. So this is supersonic motion. This is the first time humans manage to make supersonic motion, it's through the motion of the cracking. And we're gonna go and look at that in quite uh detail. But there are other things you can do, not with whip, but with chain or with whip actually. Uh, and I wanted to show you there's something else bizarre called the chain fountain. And that's something that's done with a regular chain, the one you use to turn on the light. And I would again try to do a little experiment, it's not sure to work, but it's not for a dramatic purpose, it works better if the height is uh actually higher, but also for dramatic purpose. If you want, I can put my glasses on so that look more serious and more responsible. Okay, so let's see. Let's get one and two and three. So all I'm going to do is I have a chain and I'm gonna release it. And you can see all yeah, the idea it goes, well, it's not very impressive, this one. But I have a high-speed move. I have a movie, and you'll see exactly what I mean with it. You see it's the same beaker, the same chain, and you have this chain going up, and very satisfied with myself, of course. I have my uh trademark thumbs up. Okay, so I also took the same chain and went into we have a little lab in the math department in in Oxford. We call it the uh mathematical observatory, where we have a very good high-speed camera, and I filmed that in high speed. And here you really see the effect, the chain going up. And what's amazing is that it doesn't touch the beaker, the whole thing goes and unravel, goes up and so this was popularized by Steve Mole in 2013 in a series of YouTube videos. You can really see the effect on popular science now of the arrival of YouTube. And the same year, uh John Biggins and Mark Warner from Cambridge wrote a first paper and called it the chain fountain, trying to give the first explanation for it. So that's also a very bizarre phenomena, it's quite beautiful, especially in slow motion, where you have this elevation of uh of the chain going up uh with no good reason. There is nothing pulling it. And so we want also to try to understand T and uh in the same context. So now it's time to turn and look at facts, you know, with practice, we've tried a little things. What do we know about whips? Well, one of the interesting things is that whips were with us for at least 3,000 years. If you look at this stone panel from uh Mesopotamia, Neo-Assyrian gypsum panel, you'll see that right there you have the whip used to uh direct uh horses. And through the year, whips have evolved uh technologically to rather sophisticated objects that typically have a handle, depending on the size of the handle. You can have a stock whip, a bull whip, or a snake whip, the one you you need to put in your saddle. You have a swivel, the place where you can turn, then you have a long part, the tongue, which is tapered and very flexible. And then it ends up with this part, which is called the fall, that direct the motion essentially to be flat, and always a little piece called the cracker that you can replace easily. That's the one that takes all the abuse and turns around and gets the most forces. So we think that we were the first to invent that, but some scientists in 97 uh thought that actually uh dinosaurs were way ahead of us. So when I was a kid, the apatosaurus were called the brontosaurus, and that means the the dinosaur of thunder, which was quite quite uh convenient, quite appropriate, because they did a little computation using a chain and showed that as the as the dinosaur whips its tail, the end of it reached supersonic speed, and that it might have used it for some purpose. So now you can speculate what it was. Uh and that that got a lot of people excited at the idea. Uh however, more recently, in uh four years ago, people did a more careful analysis with more like the vertebrae and all that, rather than a simplified physical model, and showed there is an acceleration but much more modest, only like 30 meters per second. So we don't know, but maybe it's unlikely that the dinosaurs were supersonic. So now what about on the science side? You play around, you can crack and all that. What is the knowledge there? Well, you have to wait until 1927 to have the first set of experiments that actually demonstrated that the the whip cracking was a supersonic motion, and that's from Zeferin Carrier, a French physicist from Toulouse. And he reasoned that in the motion of the whip as you go, as you do a snap, there is a piece that doesn't move, essentially the middle piece here, where you go around. And so if you want to do a control experiment, what you do, you replace that by a pulley, and now you have a pulley around, and all you need to do is put a weight here and let it unravel so that it accelerates with constant acceleration. So that was a very good idea, and you can imagine 1927 he actually managed to do high-speed photography. Must have been extremely complicated from an experimental point of view. But he had series of results, fascinating. He in red you see the pulley, and you see the result, the different stage of the chain or the of the string that he has going around and flapping around. And he also measured the velocity, and what he showed for the first time at that time was that the tip of the whip, or the tip of his string, was 900 meters per second, which is about three times the speed of sound in the air. Right? Speed of sound is 340 meters per second. And that's at that point, this is where the little cracker at the end goes from one side to the other side. So a few years ago, we thought, hmm, that's very interesting. And could we redo this type of experiment? And we thought that we simplify something in between the chain front end that I show you and the uh and the whip. And so what we did, we take a beaker like that and a string like this one, and we just added a weight on the string and drop the weight and see how this part moves around. Very much like the original uh experiment of Carrier. This is a movie of it. What you see, the big black uh circle is the is the beaker, and you see the motion, the chain being pulled, and this is what it looks like. That was the work with P.T. Barr, Basile, and Dominique. It was a lot of uh fun problem also to try to see that. And the interesting feature here is that it leaves also the beaker, and that means that the acceleration at one part of the chain is lower than the acceleration at the other part. It goes faster, that's why it lifts. And so you can ask why is this acceleration, extra acceleration coming from? And that's very much like the chain front, and as it goes around, this part has faster acceleration than the chain going down. And the other feature of interest is to look at what happened right at the crack, because the the chain or the or the whip is actually very curved there. And right at that point, if you look at the radius of curvature, it goes flat at some point, the radius of curvature goes to infinity. And that's very interesting because in mathematics and in physics, when you have this kind of behavior, something goes infinity, a lot of things simplify. It's a good place to study. It's called a singularity, and singularities are always uh interesting. So we're going to think about that, we come back to that. Another interesting paper was the paper uh in 1998 from German physicists, and they have this figure in the paper. Uh, a guy with kind of a, it turns out he's a Black Forest whip cracker, and they brought him in the lab with his costume and everything in order to do uh high-speed imaging. And they do they did beautiful high-speed imaging, not only looking at the the full motion of the tip, but on top of that they have something called Schlieren imaging, which captures the variation in the density of air, and that allows you to see uh supersonic crack as it's a supersonic wave emanating from the whip. And so you can exactly pinpoint when the crack happens and where it happens, which turns out to be uh important for a problem. So it is looks like that. And each frame is a hundred, about hundred uh microseconds, the whole movie is two milliseconds. And if you extract a few frames, what you'll see right at the time it cracks when the supersonic boom arises, it's right at these frames. And here I uh enhance them. This is what you see, the the supersonic wave coming out of it. If you had a bullet in with the same time of imaging, you'll see the cone, the MAC cone emanating. We'll see that a little bit later. So that's very interesting because they also had very good data. And what the data showed is that if you look at the velocity as a function of time, so that's the point here, a velocity a function of time, the acceleration of the tip of the tip, the cracker, right, is about 50,000 times the acceleration of gravity. So you get amazing acceleration, and deceleration is even faster, about 70,000. And you, since you have imaging, you can exactly know when the crack occurs. And the interesting things that they show is that the crack occurs right there, and the maximum speed at that time of the tip is twice, about twice the speed of sound, which is very strange because I just told you that the popular explanation is that a whip cracks when the tip reaches speed of sound, not twice the speed of sound. And so that's also a bit of a puzzle for us. Uh try try to uh understand that. So now we have a few facts, and for me the the basic or most interesting thing is to see how starting with a very simple motion, which is my arm one millip uh one meter per second, you can get to this tremendous acceleration and velocity. How is that possible? How does that work? What are the important effects, tapering or the way you do it, and so on. So we are part of now trying to throw estimate at that, back of the envelope computation. And the you'll find more for more than a century, actually, Mack himself did computation of this type, uh, what I call a naive computation. You you're going to decide that you know you know the motion of the whip, and that initially you it's the whole whip moves at a velocity u. So as I go like that and move, the whole thing, as it moves forward, has a velocity u. So the energy is the mass of the entire things times the velocity square divided by two. That's kinetic energy, one basic quantity. So we have the mass of the whip and the little m is the mass of the cracker. So we have one half big m, the mass of the whip of the of the tongue moving, not the not the stick, and uh starting at velocity u, so it's one half m plus little m u squared. And then you're going to postulate that energy is conserved. Yeah, there is some friction and things like that, but overall it's not a bad. But idea, if energy is conserved, then as you turn, there is a part of the whip that's not moving anymore, and the rest of the whip that goes further that goes above. So now a fraction of the whip is only moving as you do this turn. And now the whole thing goes at velocity v, that I don't know. And the velocity v depends on the position of how much is moving, that's decided by the length x, a fraction of the mass of that related, and the little mass m. And now what you do in physics is you say, well, the energy is conserved, so e is the same for both, and I can solve for the velocity as a function of x. And so I have the velocity, depending on the initial velocity, the mass, the little mass, and how much whip is actually moving. And that's that expression is not that interest important, but the interesting thing is when it reached the end, because when it reached the end, the whole the whip is not moving anymore, it's fully extended, only the little part, the cracker moves. And that's little m. Right? And so you get an expression for the maximum velocity as a function of the initial velocity times the ratio of the mass plus one. So it gives you an idea, and it gives you the basic physical mechanism of what is going on in this case in this system, that you start, you give the system energy, and that energy gets concentrated in a small and smaller part that's moving. And so the velocity has to increase since the mass decreases. However, if you start looking a little bit and say, okay, but what is the mass of the cracker? Are we talking the whole thing? What if I have a continuous whip that's just tapered, for instance? There is no little cracker, so the mass goes to zero. And that's fine, the mass goes to zero, but then the velocity of the tip goes to infinity. So, hmm, that's bizarre. And if you decide which part is the little m and the big m, you realize you can have any velocity between u and infinity, right? So said, okay, we have the basic mechanism, but maybe it doesn't really help us to get a better view of what's going on. Okay, let's put that aside, we'll go back to that. Let me turn out to our other problem, which was the beaker or the chain uh fountain. And you can do another computation that's very popular uh among physicists also. It's called the Atwood machine from the 18th century. At Wood at his idea that you could have a little machine and a pulley with two weights, and if you vary the respective weight of one and the other, uh the acceleration would lower the big weight, and that would give you a way to actually compute the acceleration of gravity by changing the mass. It's very popular, it's textbook undergraduate physics the way you do that. So, what is the computation? What is the back of the envelope computation? Well, you need an envelope, and so I here is the computation of the envelope. Uh one of the theorem in mathematics is that every theorem is a back of the envelope computation. You just need a big envelope, big enough envelope. So in this case, you need a small envelope, and what you do, you essentially you say that the you have conservation of linear momentum, that the force acting in one side on one at the acceleration is balanced by the other one. Right? So you have two equations of motion, one for the little mass and one for the big mass. So conservation of linear momentum, you can get rid of the tension in between, that's t, and you get easily the acceleration as a function of the big mass and the little mass times acceleration of gravity. And you can see that if though if they are the same, big m and little m, you have zero acceleration as you would expect. Now, what happens if I get rid of the little mass? I, you know, you're not supposed to do that, but why not? Right? Does it still work? Well, the big mass accelerates at acceleration of gravity. It's just falling, nothing big. Well, that's strange because when I did the experiment with the B cut, this is not what happened. Right? The I had an acceleration, but without little mass at the end, you chain has no mass hanging there, the whole change was moving up. And you see that it's one of the problems with this computation in the previous one, is that you assume you know perfectly what the motion is and you just solve for one quantity. And so it's naive computation, gives you some intuition, but you have to be very careful about the assumption that you actually make. So, what about this chain font then? What is the standard explanation for that? Well, the idea there is that if you look at two of the beads in the chain, they're connected by a little bar, and one of the beads, like the one on the left side, is on a substrate, is on top of the other pack. And the one on the right is being lifted by the chain at any given time, right? So it's just at the time where it's being lifted. And as it's being lifted, what happens is that the rotation of that gives you an extra kick that lifts you. And that seems a little counterintuitive, but if you ever done any skateboard, you recognize that's exactly the only effect. So only is uh the only effect or the only move. It's named after Alan Gelfan. Uh he was a skateboarder that introduced this motion. And the idea is that this trick is that as you go along, you rotate the board, and that rotation provides an extra kick in the back and lift the whole thing. And I'm sure you've seen that. You look at skateboarders, they do that all the time with uh with great expertise. And so you have you might have the same effect on the chain, or the the chain, you have the two bead connected. If you lift one, the other one is going to be pushed out by an extra force, like that, but because of the rotation in the balance. So another way to look at that is uh a delightful experiment by uh my friend Andy Runa in Cornell. And what he did was very interesting. He took a ladder made of steps at different at different angles, two ladder exactly the same or symmetric with respect, and he let one fall on a table and the other one fall freely. And so the question is what's the effect of the table on the motion of the ladder? Well, it's very strange because there is no reason if something falls on a table, it changes. Except if you have the same effect, as it falls on the table, the each step in the ladder would accelerate and provide a force that pulls the ladder. And so you did this beautiful experiment. You have a free-falling one on the right and the table on the left. And what you see, they started exactly at the same point, is that each of these steps is pulling the ladder down and accelerating. So the one on the left goes faster than gravity by this effect. You have an acceleration due to the shock, which is quite surprisingly. Mechanics is infinitely surprising. Okay, so we have all these uh effects, some interesting, some strange, and we really want to make sense of that. And now we get to the next step, which is to formulate uh what in the trade we call a toy model, because we like to play with it, not just the toys that I show you. A toy model is a minimal uh mathematical model that has all the uh elements of that. You see, in the previous uh uh computation I did, I assume I knew everything about the motion, and in that case, whether you decide that linear momentum or energy is conserved, you get different answers because they cannot be conserved for the motion ahead. In reality, you really have to solve for the motion and see what the consequence are there. But solving for the motion is difficult because this is a rather complicated structure. So you want to idealize it, you want to model it. And the first thing I'm going to say is that, well, the motion is mostly in the plane, as I crack it, even if I go overhead, it's motion in the plane. So I'm going to look at a rod, a string in one dimension, uh in two dimensions in the plane. So a one-dimensional continuum in the plane. Now, this object is right away much more complicated than what we've seen, because now I look at each point, x, y, and I parameterize it with the lengths, the arc length from, let's say, the swivel up to the point that it reaches. So the position of, if I paint a little red dot like here, like I did on the on the slide, this position of x and y change as a function of time. But each dot change also. So x and y are not both function of the arc length, that gives me the position, and time. So it's a function of two arguments. But you can then express essentially f equals ma with this continuum, and you get uh you get the first set of equations, which is the balance of linear momentum, which tells you that an acceleration of a little part is balanced by the forces that are inside, uh that it feels inside the material. And that's these two equations, F and G, are associated with the force of inside the material that pushes it. Now you have another equation, which is the equation for the angular momentum, which is also important. What you need now, if I look at a little piece of my material, it can also change its orientation. That provides a torque, and the torque is related to angular momentum, so I need also to balance that. That gives me a second uh third equation, and that gives me three equations for that. Don't worry about equations. T's are actually very complicated equations. T's are three non-linear partial differential equations. So when I started this Gresham job, I was told don't show differential equations because this is too complicated. And they were talking about ordinary differential equations, which is the baby version of this. T's as as complicated equation as you can get. And they are complicated because they describe extremely rich all the possible motion of a chain or a rod in in the plane, right? So they have a lot of solutions, this extremely rich set of equations. So this is a minimal model because now we can solve for both velocity and the shape at the same time if we know to solve this equation. So forget a moment of this equation. Now I have my back of trick of methods from applied mathematics, and I know how to get that, uh, extract information of that. And the first result is that loops exist. They are solutions of the equation, and actually it's more or less something that Euler knew about. And what I mean by that is that if I look at a uh a loop like this one, I know that these equations support the same type of solution on an infinite domain, things that go to zero on both sides. And more important, the loop can have any height, and I have also a nice little formula for it that is not important for us, but I can actually write the solution exactly for that. Now, that's a loop being static. The second result, and a bit of a miracle, is that any of this loop can be made to travel. And that came more recently, this type of idea of looking now at a dynamical solution. That means if I start with something like that on an infinite domain with no variation of tapering, there's a solution that makes it roll further. And you're gonna see what I mean by that. This is the solution, and I can make it travel depending on the initial condition at any velocity. So an infinite light, a little loop like that can travel. It's part of the set of solutions of this. And why is that interesting? Well, for a problem, it's not that interesting because it's constant velocity. So in this case, if I send a loop with a velocity c, it would just go velocity c forever, if there is no friction or anything like that. However, the whip is stapled. And now we can look at the effect of the tapering on the whip, and and that becomes much more interesting. And now T's loop now is stapled, and you have to now solve the equation for T's problem where the mass is actually changing, the radius is changing as it goes along. And that's that's a problem that hadn't been solved before, and I did it with uh Tyler Macmillan, who was my uh PhD student at the time. And it's very much like a problem, uh like a wave coming ashore. And if the beach is a little slanted as it comes around, you have a variation due to the fact that the depth change. So it's part of this type of problem where you have propagation of a wave on a changing medium, and there are a lot of nice applications. But in the context of this problem, it hadn't been done, so we looked at that and we solve it. And the problem is the following, going back to energy. Let's think about energy for a second. It says the physicists tell us that uh the the motion is completely confined, but if I have a loop, there's plenty of ways I can conserve energy. The loop, there is less mass traveling, it's true, but the loop can get tighter and have more elastic energy by being more curved. That would conserve the and go at constant velocity, that would conserve the energy. The loop could get bigger as it travels, because there is more mass more mass moving, it is still a solution at constant velocity, or the loop could stay the same and increase in velocity. You don't know that a priori. Nothing is telling you what happens, and you have to do the computation. And what the computation showed us is that essentially there's small variation of height, but the loop actually accelerates but remains the same height essentially. And you can now, as a function of the tapering, you can actually find exactly what the velocity is in the system. Of course, you're gonna tell me, well, wait a second, this is not an infinite rod on which a loop travel, no, it's a finite piece, but we have this effect. And the other objection would be well, if I'm if I if I do it well enough, I can take and I can do it also, I I could take a non-taped uh whip and make it crack. It provides a little bit more energy and all that, but if you know what you're doing, you could also make it cry. So tapering is not the whole story, and you want to go to the next step. Let's see, in more realistic conditions, now uh what happens exactly in the system. The good thing is that you still have all the equations, but as far as doing them analytically, pure mathematics, getting formula and all that, that's probably the best you could do, is what I showed you. What you can do, however, is develop numerical methods to solve this equation. That's also part of the bag of tricks that we have. And using these methods, you can start with an initial loop, as I show you, and now look at the effect of having an open end at the end rather than an infinite one. And then you can vary tapering, you can vary the length, you can vary the force, the initial velocity, and you can exactly point out what are the different effects that matters in order to get an acceleration at the tip. And you can you can tabulate and you can show that all these effects are somewhat important, both tapering, uh, force, initial motion, and all that, exactly like you would if you practice the whip, you would have the right coordination to send the right wave and pull it, put the right force at the right time, and you get very quick acceleration. And since you have the whole motion, the whole motion of the whip as it as it goes around, you can now look at the velocity at each point. And the velocity, look at the tip speed, for instance, follows very much the type of experiment that you saw. And so if now you look at uh the chain experiment on the beaker, the one that's nap, I don't know if you can see it well on the screen, there is a yellow curve, the the black dot is the experiment, and the yellow curve is the numerical solution of the same type of equation that I show you. And as you see, the the fit is actually extraordinary. It's not a fit, there is no free parameter. This is just a perfect prediction of the motion of the chain as you go along. And because you have that, you can now know from the numerical solution, the one the computer gives you, you have all the possible information. In particular, you can look at the curvature. And when you look at the curvature, you can look at the place where the radius is maximal, uh, the so the smallest possible curvature, the way the place where, as it unfolds, the chain is most curved. And its inverse is the radius, is the best-fitting circle at that point. And as it goes through this transition, there is a critical time where it becomes flat, and that means the radius of curvature goes to infinity. And that's what is called a singularity, and that's where the snap occurs. And this type of singularity is very interesting, it hasn't been completely resolved. It's a good place to study, but exactly how it goes to infinity is still a bit of an open, open question for us. But so far, all I've shown you, it was a toy model, did not include the motion, but no interaction with the air. And the crack sound that we hear is really a change in air pressure. So I want to go back to uh supersonic motion, and as you can see in Arizona, not only we have WIP, we have road runners, and maybe they're fast enough to be supersonic, who knows? So the first thing that I mentioned was that little puzzle, a little surprise of uh the tip loop, uh the velocity of the tip as it cracks. So remember that when you look at the data, you can show that the whip cracks when the tip reached twice the speed of sound. At first that seems a little bizarre to us, and then we realize the following is that in a motion where you have a loop, the motion of the loop, the velocity of the loop, is different than the velocity of the point on the loop. So let me give you a much simpler example. If I take a bicycle wheel and I roll it along a wall at a velocity V, you can ask what is the velocity of the material point right at the top of the wheel? Well, the velocity at the top of the wheel must be greater because that's why it goes in front of the wheel. Right? And if I roll it at constant velocity, you can show by nice geometric argument that the velocity of that point at the top is always twice the velocity of the wheel. Perfect. And that's true for any shape, actually. If you have a shape that travels at constant velocity and like that, the top will be always twice. So it's not a property of the physics, it's really a property of the geometry. So if that shape doesn't change, the loop goes at velocity c and the top of the loop goes at velocity 2c. So as it unfolds, you have C prime, it goes faster and faster because of tapering or other external extra tension or things like that, up to the point where the loop itself, now the loop itself, so you have a loop, and the loop itself reach the velocity of the sound, but the tip that's is being protected by the loop. The tip does not go against the air. So it's the loop that breaks the sound barrier, not the tip that breaks the sound barrier. The tip comes after and unfolds. And the way to see that is to know back to the data when you realize that you say, hmm, let's see if we can actually see that on the data. And if you look at the experiment where you see it is supersonic motion, here's the speed C, and you see this is where the supersonic shock is being created, the supersonic wave, it's the loop, not the top of the tip that at velocity 2c as it unfolds. And then you have further uh uh expansion. Now, supersonic motion is very interesting, very difficult, because you have to understand the physics of air and compression and jump and all that in very extreme conditions. But there are very simple things you can do. Uh, one of the simplest constructions is the MAC cone. So if you have a motion in the air, just by friction in the air, uh the that that point is going to uh send sound wave, just like you hear something going in inside a room if you throw it fast enough, is the sound that the air makes at its expense. But if that point is going supersonic, the sound waves do not have time to expand fully before they all coalesce on something called an envelope. And so if I have this sound wave emanating constantly from that moving point due to interaction with the air, if the velocity of my point is faster than the speed of sound, they all accumulate with an opening, the Mac Con opening, which is given, the sign of that is given by the ratio of the velocity of the sound by the velocity of the object. And this is what you see in Schlieren imaging for a plane or for a for a supersonic bullet or something like that. So for a plane, you might experience that you sometimes hear more than one boom. So the supersonic sound, the supersonic boom, is not when the plane gets past the sound barrier. It's something that's constantly emitted by ulti-sound wave gathering. So at that point, if you sit right at the intersection there, it's when that red line crosses you that you hear the sound. But a plane is an object of finite size, and so there are a lot of different waves being emitted by different pieces of the plane, and you have one main one that's from the tip of the plane that creates the first one, and you have a change in air pressure, and then you have the opposite change in air pressure at the end of the plane. You have a change. And so that's why you can hear a double double boom or multiple boom as supersonic plane goes by. But there is a huge set theory about uh supersonic motion, or bullet or planes or space and all that. But almost all of them, as far as I know, all of them were for uh rectilinear motion, because usually people, plane goes go straight, or bullet goes straight. And there is no equivalent what if the motion actually uh is not straight but circular, just like is as the whip cracks, it unfolds. But it's not too hard to do. You can play the same geometric trick and you can have a point. Now think about the the cracker, the loop as it unfolds. Now it rotates rather than going straight, so it follows more or less a piece of a circle, and that piece of a circle also emanates waves, and you see now that you have a curved macode depending on the trajectory, and you can now compare that to the data. And so by putting all these um different pieces of physics and mathematics and all that, you start having a very good picture of all the different effects and a good understanding of whip cracking. Uh that's that's quite nice. Of course, you can say, well, that seems a little bit frivolous, putting all your effort into uh into that kind of problem. What is it good for? Uh well, first I want to say that it's not a question that people ask to people who study black hole or or the origin of the universe and so on. But we do have a responsibility. What is it good for? So this example is of course very uh catchy because it's very extreme, but it's actually part of a whole body of problem for which we develop methods. It turns out that this one is one that that captivates the attention. So we get to the problem or the question, so what? So what else can you do with that in a way? Just explain the cracking of the whip or chain, this this is good, but that's not what we pay you for, right? We pay you to help society. Well, there are plenty of other problems when a cable snaps or something like that, but if you look at the smaller scale, there are microorganisms, for instance, any kind of flagellate organism that uh beat the flagellum and in doing so provide linear momentum that's expelled for the motion. So this one is not a little uh space lunar module. If you look at the scale, this is three micron, it's a very uh nice uh uh organism called a coanoflagellate that has this specific tail that flips. And these are very close in some aspects to actual sperm and the flagellum of the sperm. And indeed, sperm do create this whipping motion by making loops and pushing them in order to propagate. And all of a sudden, you went from the whip to understanding something about motility of the sperm, which is a very important problem view both on physiology and biology. And we've applied the same type of problem techniques. Again, you completely have to refine your model because nautics operate in very viscous environments, so you have to take into account the interaction with the fluid, the viscosity on one, the other one, and so on. But using that we manage with uh Emman Gaffney, which is in Oxford, and and uh Hermes Gadea, who is now in Bristol, uh, to explain some effects that are observed in in human sperm uh related to uh what is called the counterband phenomena. Another cool example uh that uh I that that I found is uh prey catching in octopus. And here what you have is the octopus very quickly unfolding and doing this nap in order to catch a prey that is uh indicated. And that as you can see from the scale, this this happens in about uh 400 milliseconds. And one of the uh last examples that you that you uh can see also, like nice application, is a much bigger scale. Uh for instance, the space tether. So satellites have these tethers, which are long rods that uh uh have a weight at the end, and they use that either to stabilize or change their orbits. It's the the the mechanics of that is very nice in terms of orbital motion. But there is one problem is that these tethers sometimes have to be retracted. And when you retract the tethers, all the shape and the motion gets concentrated as you retract the cable, you call you coil the cable in smaller and smaller portion. And now in space there's very little friction, so that you don't have viscosity and all that, and so what happens is that as you coil it, the motion of the end gets wider and wider. Very much like when you eat spaghetti and you slurp a spaghetti, and what happens is that all the energy that you do for the velocity of slurping is being transferred, or the energy is being transferred in a smaller and smaller portion. And if you have that with uh pasta saus, you know what happens. It oscillates and it splashes on you. And that's exactly the same phenomena that happens. I've never worked on space tethers, but they're very interesting because in in space the problem is that you it's very hard to uh to stop things from happening. The motion of objects in space is very much uh inertial, that means not damp. There is very little dissipation that you can, but and that creates uh other problem. So I hope that I've given you a global picture, starting with a very simple problem and bringing up to a more complicated one, but one for which we have no answers and uh different application to the rest of the world on that. Thank you very much.