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Lecture 5: The quest to predict the future

Scientists try to predict the future by finding rules in nature, and the language of these rules is mathematics! In this lecture Marcus uses fish and cannon fire to demonstrate how, when a pattern is discovered in the physical world, mathematicians can describe what will happen by using an equation. So, for example, you can predict how far something or someone will travel when fired from a cannon if you know their weight and the angle of the cannon. Surprising things happen in nature though, which means that sometimes mathematicians' predictions are wrong and the opposite occurs. And, as Marcus explains, the weather is difficult to predict because so many factors have to be taken into account - as well as seasonal changes, the smallest thing can have a dramatic effect. If you want to become a weather forecaster you'll have to get to understand Navier-Stokes equations and that butterflies may cause hurricanes!

Further and Deeper

Try this out

Build your own boomerang then test out ways to change its flight path.

Boomer it better (PDF Document 120KB - new window)

 

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More Maths

Download this pdf to discover the answers to such questions as: what is chaos?, why does it occur?, and what can we do about it?

Why does it always rain on me? (PDF Document 90KB - new window)

 

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Lecture 5 transcript

The quest to predict the future (PDF Document 160KB - new window)

 

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Web links

Watch this fascinating video on the effect of vibrations on a cornstarch solution. Towards the end are some very surprising results.

Chaotic fluids (external link - new window)

 

Play this gravity game and see how a small change in the initial conditions can create a large difference in the final trajectory.

Gravity game (external link - new window) (Requires Shockwave Player 10.1)

 

Confused about swerving free kicks in football? This BBC magazine article explains how mathematics is the key to understanding them.

Free kicks and mathematics (external link - new window)

 

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Video clip - Chaotic magnets

Marcus paints chaos theory with magnets and shows why weather forecasts can easily be wrong.

(Press play to start the video.)

This exciting, interactive site has been designed using Flash 8. In order to get the full experience, you will need to download Adobe Flash version 8.

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Welcome back to the Number Mysteries and our quest to predict the future. Before the break I asked you to predict the behaviour of this double pendulum. I wanted to know whether you thought the last rotation would be anti-clockwise or clockwise around here.

Now some things are very simple to predict, the future. But look at this one, this is much harder to predict exactly. Even when it's going to stop going through the top, seems to come quite stable then it goes round, seems to be going anti-clockwise, anti-clockwise, so you think it's going to end anti-clockwise. Oh no, it's decided to go clockwise. Has it stopped? Oh no, no, it keeps going. So you think how an earth could you predict this? It seems incredibly erratic behaviour. I've just made it a little bit more complicated, put two things. Has it stopped there, was that the last swing of it? So, did anyone notice whether it anti-clockwise or clockwise last? Yes, what did it go? Anti-clockwise though, but this is so erratic. In fact sometimes it's so surprising it starts kicking again, but I think that is its last go. So, it's incredibly difficult to predict this. This is so erratic that we call it chaotic.

In fact, chaos theory was discovered over a hundred years ago by a French mathematician called Henri Poincar. He was interested in understanding whether the solar system was stable or not. No one knew whether the planets, as they went on, they might be very stable, but suddenly they might fly off into outer space. Was the thing stable or not? Well it's pretty crucial stuff to know whether the planet we're living on is actually going to be stable and not fly off, so Poincar began to investigate. Now, mathematically, two bodies is very simple. If I take the earth and the sun, that is very easy to do. It's very periodic; just repeats itself again and again and again, when I do that. But if you add just one more body, a third body, for example, the sun, the earth and the moon, things become much more interesting. I'd like a volunteer to try and explore what happens in the universe when we've got three things in it. Let's have you up. OK, why don't you come here? So this is our little Poincar, we're going to go over here not from this universe, to our little universe over here. OK, so what's your name, sir?

Harry.

Harry is going to explore a little universe over on the floor here. OK, you kneel down. You can have a look at it. Don't pitch yourself in, gravity is sucking him in, I think. Why don't you come round here a little bit, so you're next to me? What I've got here are three magnets, Harry. They're different colours: we've got a red magnet, a green magnet and a blue magnet. And the magnets attract the pendulum, so this is a bit like three planets and this is like some sort of asteroid which is going to fly around. So, yes, I'm going to have some paints coming out of here in a minute and what I want you to do is predict if I let it off from here, which one of the three colours you think it's going to land on. Which one, do you want to make a guess, if I launch it here?

Blue

Blue, OK, you're going to go for blue. So Tim, if you want to remove the paint and we'll get Harry, once the paint's removed, put your hand on it. OK, let it go. So this is now going to map out a little trajectory for us, this is what our asteroid is doing. OK, there's seems to be, wow. I think it likes the blue, but it's a completely wild path where it's going. Is it going to end up blue? Seems to be liking blue, oh no, it's gone green, blue, blue. Oh fantastic, here's somebody who can predict the future. OK, Tim if you want to take that one off? Absolutely amazing. Well we've got a budding mathematician in Harry here. Though what I want you to do now, and this is what Poincar discovered, is that I want you to try and repeat what you just did. So Tim is going to put some more colour on, a different colour, and we're going to see, we know where the path started, we know what direction it went in. Let's see if you can repeat exactly the same path in this new colour. So Tim, if you could give me my new colour, what colour have we got now? Black, OK, let's see if you can, Tim can you help us take the thing off, I don't like getting my hands messy I'm a mathematician. So see Harry if you can predict, get exactly the same path, so we know where it's starting, let it off. That's pretty good, yes. It doesn't look like the same path at all, but will it end up in the same place? Let's see, it started around here, green no, red, red, how amazing, it's started to go nearly exactly the same place, but the path was completely different. Tim, do you want to take that away? It ended up in a completely different magnet, and this is what Poincar discovered. With the solar system, this is a bit like a little solar system, make a small change in the solar system and things can behave completely differently and go in a different direction. Well, let's give Harry a big round of applause.

So we live in a clockwork universe described by Newton's laws of motion, but what Poincar discovered is that if you change the initial conditions, just slightly, then the solar system can fly off in a completely different direction.

Computers have actually helped us to understand a little bit more about these equations of chaos. In fact, I've got over here, a computer print out which helps me to predict what was going to happen in this magnetic pendulum. So I have three magnets down here. This, the computer, calculated what will happen to the pendulum. So if I'm over a red spot and I release the pendulum from there, it means the magnet will end up at the red moment, sorry pendulum will end up at the red magnet. But if I release it over a blue spot, then that says it will up with a blue magnet. So here the regions are very predictable. I can move it around a little bit and it will still end up at a red magnet, same here with the blue. You see where Harry set off his pendulum was right over here. And you might be over a little blue spot and go here, but if you shift it very slightly you're over a red or a green and it can go somewhere very different. So, it's important to stress that the equations here, if I started in exactly the same position, it will go to exactly the same magnet. The point about chaos theory is that if I change it very slightly then it can veer off in a completely different direction. Now this chaos makes it very difficult to predict the future. And chaos is everywhere. The weather, for example, is chaotic. It's controlled by very similar equations to the ones that are controlling this pendulum. So, for example, there are regions in the weather, say if I'm here, well this is quite predictable. I change the weather a little bit, this might be a desert or something where it's very hot, if I change it a little bit, it's still going to be hot the next day. And this might be the Antarctic, or something where it's cold, change the conditions a little bit, it's still going to be cold. But we're kind of living, really, somewhere over here, where the weather is incredibly unpredictable. And that's why weather forecasts, you know, three-day weather forecast is about the most we can do. So, for example, you might start off over a blue and predict that it's going to be cold tomorrow, but you move these conditions very slightly, just might be a small decimal place, it might move to the red and suddenly be hot tomorrow.

Welcome back to the Number Mysteries and our quest to predict the future. Before the break I asked you to predict the behaviour of this double pendulum. I wanted to know whether you thought the last rotation would be anti-clockwise or clockwise around here.

Now some things are very simple to predict, the future. But look at this one, this is much harder to predict exactly. Even when it's going to stop going through the top, seems to come quite stable then it goes round, seems to be going anti-clockwise, anti-clockwise, so you think it's going to end anti-clockwise. Oh no, it's decided to go clockwise. Has it stopped? Oh no, no, it keeps going. So you think how an earth could you predict this? It seems incredibly erratic behaviour. I've just made it a little bit more complicated, put two things. Has it stopped there, was that the last swing of it? So, did anyone notice whether it anti-clockwise or clockwise last? Yes, what did it go? Anti-clockwise though, but this is so erratic. In fact sometimes it's so surprising it starts kicking again, but I think that is its last go. So, it's incredibly difficult to predict this. This is so erratic that we call it chaotic.

In fact, chaos theory was discovered over a hundred years ago by a French mathematician called Henri Poincar. He was interested in understanding whether the solar system was stable or not. No one knew whether the planets, as they went on, they might be very stable, but suddenly they might fly off into outer space. Was the thing stable or not? Well it's pretty crucial stuff to know whether the planet we're living on is actually going to be stable and not fly off, so Poincar began to investigate. Now, mathematically, two bodies is very simple. If I take the earth and the sun, that is very easy to do. It's very periodic; just repeats itself again and again and again, when I do that. But if you add just one more body, a third body, for example, the sun, the earth and the moon, things become much more interesting. I'd like a volunteer to try and explore what happens in the universe when we've got three things in it. Let's have you up. OK, why don't you come here? So this is our little Poincar, we're going to go over here not from this universe, to our little universe over here. OK, so what's your name, sir?

Harry.

Harry is going to explore a little universe over on the floor here. OK, you kneel down. You can have a look at it. Don't pitch yourself in, gravity is sucking him in, I think. Why don't you come round here a little bit, so you're next to me? What I've got here are three magnets, Harry. They're different colours: we've got a red magnet, a green magnet and a blue magnet. And the magnets attract the pendulum, so this is a bit like three planets and this is like some sort of asteroid which is going to fly around. So, yes, I'm going to have some paints coming out of here in a minute and what I want you to do is predict if I let it off from here, which one of the three colours you think it's going to land on. Which one, do you want to make a guess, if I launch it here?

Blue

Blue, OK, you're going to go for blue. So Tim, if you want to remove the paint and we'll get Harry, once the paint's removed, put your hand on it. OK, let it go. So this is now going to map out a little trajectory for us, this is what our asteroid is doing. OK, there's seems to be, wow. I think it likes the blue, but it's a completely wild path where it's going. Is it going to end up blue? Seems to be liking blue, oh no, it's gone green, blue, blue. Oh fantastic, here's somebody who can predict the future. OK, Tim if you want to take that one off? Absolutely amazing. Well we've got a budding mathematician in Harry here. Though what I want you to do now, and this is what Poincar discovered, is that I want you to try and repeat what you just did. So Tim is going to put some more colour on, a different colour, and we're going to see, we know where the path started, we know what direction it went in. Let's see if you can repeat exactly the same path in this new colour. So Tim, if you could give me my new colour, what colour have we got now? Black, OK, let's see if you can, Tim can you help us take the thing off, I don't like getting my hands messy I'm a mathematician. So see Harry if you can predict, get exactly the same path, so we know where it's starting, let it off. That's pretty good, yes. It doesn't look like the same path at all, but will it end up in the same place? Let's see, it started around here, green no, red, red, how amazing, it's started to go nearly exactly the same place, but the path was completely different. Tim, do you want to take that away? It ended up in a completely different magnet, and this is what Poincar discovered. With the solar system, this is a bit like a little solar system, make a small change in the solar system and things can behave completely differently and go in a different direction. Well, let's give Harry a big round of applause.

So we live in a clockwork universe described by Newton's laws of motion, but what Poincar discovered is that if you change the initial conditions, just slightly, then the solar system can fly off in a completely different direction.

Computers have actually helped us to understand a little bit more about these equations of chaos. In fact, I've got over here, a computer print out which helps me to predict what was going to happen in this magnetic pendulum. So I have three magnets down here. This, the computer, calculated what will happen to the pendulum. So if I'm over a red spot and I release the pendulum from there, it means the magnet will end up at the red moment, sorry pendulum will end up at the red magnet. But if I release it over a blue spot, then that says it will up with a blue magnet. So here the regions are very predictable. I can move it around a little bit and it will still end up at a red magnet, same here with the blue. You see where Harry set off his pendulum was right over here. And you might be over a little blue spot and go here, but if you shift it very slightly you're over a red or a green and it can go somewhere very different. So, it's important to stress that the equations here, if I started in exactly the same position, it will go to exactly the same magnet. The point about chaos theory is that if I change it very slightly then it can veer off in a completely different direction. Now this chaos makes it very difficult to predict the future. And chaos is everywhere. The weather, for example, is chaotic. It's controlled by very similar equations to the ones that are controlling this pendulum. So, for example, there are regions in the weather, say if I'm here, well this is quite predictable. I change the weather a little bit, this might be a desert or something where it's very hot, if I change it a little bit, it's still going to be hot the next day. And this might be the Antarctic, or something where it's cold, change the conditions a little bit, it's still going to be cold. But we're kind of living, really, somewhere over here, where the weather is incredibly unpredictable. And that's why weather forecasts, you know, three-day weather forecast is about the most we can do. So, for example, you might start off over a blue and predict that it's going to be cold tomorrow, but you move these conditions very slightly, just might be a small decimal place, it might move to the red and suddenly be hot tomorrow.