 Well, welcome to the department of continuing education. I'm Marcus DeSotoie and I'm a professor of mathematics here at the University of Oxford, and I'm also the Simone Professor for the Public Understanding of Science, which I took over from Richard Dawkins. And the professorship is shared half with the department of your subject, so mathematics, but half with continuing education, because of course that's one of the goals of this department, is really to spread the amazing work that's done in science here in Oxford and beyond, and to engage as many people as possible in this wonderful world. It's interesting, I'm a mathematician and I think mathematics is an interesting choice for this professorship, because I think mathematics actually does underpin so many of the other sciences, it is really the language of science. But it's intriguing, when I'm at a party and people ask me what I do, and I say, I kind of look forward and dread this question, because when I say, oh, I'm a mathematician, you can sort of see the look of horror descend on most people's faces, and they sort of start backing away, and they find their glasses become very empty, but I'm very persistent, so I chase after them, and I try to explain to them that, actually no, no, mathematicians were a very misunderstood breed, and I think that most people have this perception that what I'm doing in my office in the master parlor just across the road is sort of doing long division to lots of decimal places, and that's the higher research, the further you go down there, and truly I've been put out of a job now as a computer, but I try to explain to them that mathematics is something very different. Mathematics is really the science of pattern searching. It's about trying to find the pattern and the logic and the order in this kind of chaotic mess of the world around us, and trying to read messages into that, and I think it is an incredibly powerful language for actually understanding where we've come from, understanding what happened in the past, but also perhaps more importantly in predicting what could happen in the future. And what I want to do for you is to take you through some of the stories of how powerful maths is at making predictions, but also the limitations of this language for actually trying to look into the future. I think that the world is faced with huge kind of problems of things like climate change, population growth, virus control, what happens if a virus spreads, and mathematics is a very, I think, ability to be able to read the data that's gone and be able to work out what possibly could happen next is potentially something that could have a massive impact if we really understand these patterns on the future. In fact, next year interestingly, 2013 is going to be a world maths year, but maths for the earth, the way that mathematics can perhaps answer some of these really big problems that face us. So as I said, maths is a subject of looking for patterns, and so I'm going to kick off with just warming you up mathematically, so I've got a few challenges for you to see how good you are at pattern searching, and so there are these kind of challenges that you probably remember from school a little bit, where you get given a sequence of numbers, and what you've got to do is to try and read some sort of structure and logic pattern behind them, and then next step is to be able to tell where it's going to go next. So some of you may have a maths degree. You're not allowed to play on the first couple of these, okay? But if you haven't got a maths degree, let's say the first sequence, one, three, six, ten, fifteen, twenty-one, twenty-eight. What's the next number in that sequence? Thirty-six. Thirty-six is the next number in that sequence. How do you get that? Well, what you do is you add the numbers together. So one plus two, plus three, plus four, and that gives you the next sequence. In fact, we call these the triangular numbers because you can view them very graphically if you take a triangle and you add an extra layer on each time. It's the number of stones you need to make larger and larger triangles. Now interestingly, we know a lot about this sequence. For example, there's actually a formula which will tell you what, say, the hundredth triangular number is without having to do the hard work of, I mean, I could add up the numbers from one, two, three, plus four, one plus five, all the way out to a hundred. But mathematicians are rubbish at arithmetic, actually. So I'd probably make a mistake somewhere on the line. And actually, there's a very cunning formula. If you put two of these triangles together, you'll get a rectangle. And currently, things in a rectangle is really easy. So you just half the number of the stones in a rectangle and you get a formula for the triangular numbers. And that will tell you the hundredth triangular number, the millionth triangular number, without having to do the hard work of adding up all the numbers. I mean, mathematicians love looking for patterns, but we're also very lazy at heart. So we love these kind of shortcut formulas to find these numbers. The next sequence, if you've read the Da Vinci Code, you're not allowed to play on this one. If you've got a master way, you're not allowed to play on this one. But if you haven't, then what's the next number in this sequence? Thirty-four. You answer the second sequence first. That's right. You're jumping ahead. So yeah, the next number in this sequence is thirty-four. There's a very famous sequence of numbers called the Fibonacci numbers. So if you haven't got the pattern, what you do is you add the two previous numbers together to get the next one in the sequence. So thirteen and twenty-one is thirty-four. Twenty-one and thirty-four is fifty-five. And these are probably nature's favorite numbers. You find them all over the natural world. It's obviously amazing in sort of pine cones and pineapples, flowers. If you take a flower and you count the number of petals invariably it's a number in the Fibonacci sequence. Sometimes you get double the number because you get sort of two copies of the flower on top of each other. And if it isn't a number in the Fibonacci sequence, that's because the petals fallen off your flower. Which is how mathematicians get around exceptions. Interestingly, these numbers actually shouldn't be called the Fibonacci numbers. They're named after this Italian mathematician Fibonacci from Pisa, 12th, 13th century. He discovered this connection to nature. Apparently they were discovered some time earlier in India by musicians and poets because they can also be used to count the number of rhythms you can get with long and short beats. So it's interesting that artists actually discovered these before the scientists did. But they come up all over the place. And again we actually have a formula for these numbers. It's a slightly more complicated formula involving the golden ratio, which is the proportions that we like in art and comes up a lot in nature as well. There are quite a few mysteries about these numbers as well. But you can use this formula again to tell what the hundredth Fibonacci number is without having to add up all the numbers from one to another. The pair is all the way up to the hundredth number. So it's again a very powerful formula which will tell you quite a lot, but there's still some mysteries about these numbers. Okay, the next sequence. One, two, four, eight, sixteen. 32. Well actually 31 is the next answer in this sequence. Two of course is the obvious one. You're just doubling the numbers. But let me tell you why 31 is equally good an answer to this sequence. Because it's describing something which starts off as if it's doubling. But then, and this is the fun thing in math, suddenly you think there's a pattern and it can go in a completely different direction. I mean if everything was sort of too predictable it would be boring. And the fun thing about mathematics is there's this kind of curious balance between things that seem to have a lot of structure and that's what it arises in. So I call these the circle division numbers. So what are the circle division numbers? Well, if I take a circle and I put one point on that circle I've just got one region. But if I add another point and join those up I've got two regions. Okay, so two dots, two regions. If I put a third dot, I join the dots up, you get a triangle in the middle, three regions around the outside, so that's four regions. Okay, I just put another dot. So four in the middle, four in the outside, eight regions. So you think, okay, I'm starting to spot the pattern here. If you add another dot you just double the number of regions. Seems pretty convincing and sure enough you do another one. You take five dots and you join all of those up. You get this five pointed star, the pentagon. You count the number of regions, 16. You think, okay, I really know, I've spotted the pattern and then this surprise happens. Because if you put six dots on, and I challenge you on a bit of paper later, however you arrange them, you cannot get more than 31 regions out of this, rather surprising. So you think you know what's happening, but there's a warning here. Patterns might go in sort of strange direction. It can lead you a little bit astray. We've had a formula which will tell you the number of regions. It's a little bit more complicated than the other ones. You have to take the number of points on the circle, raise that to the power four, then take away a cubic and blah, blah, blah, blah. But amazingly this will tell you the number of regions that you have as you add more and more dots on. Okay, the next sequence. If you've got a master degree, you can play now. So 2, 9, 10, 11, 13, 16. A little bit more challenging. I thought, oh, this is easy. I know Fibonacci numbers and stuff. Anyone got a master degree here? You're dead or not. Yes. Well, if you could get the 26th of the next number in this sequence, I recommend you buy a lottery ticket next week because these were, in fact, the lottery ticket winnings for the 28th of September. So yeah, another warning here. Not everything does have patterns. Some things have genuine randomness in this, and that's often important as well when you've got a lot of data. We live in a world of huge, big data, and sometimes being able to know when something is just random or when there is a signal inside there is really important. So being able to actually recognize the signature of when something is random is important. I'm pushing, I don't have a formula for these. If I did, I would not be here now. I'd be on some tropical island enjoying myself. Okay, so the next sequence. 2, 3, 5, 7, 11, 13, 17, 19, 23. So these are the prime numbers. These are the prime numbers, I think, which are the most central to the whole of mathematics because they are the ones I spend a lot of time as a research mathematician here trying to understand because they're also some of the biggest mysteries are about these numbers. We really don't understand them at all. But they are so important because they really are the building blocks of my subject. If you take a number like 105, that's cleaning up prime, so this is all by 5. Divide by 5, I get down to 21 times 5. 21 isn't prime, I can divide that to 3 times 7 times 5. But now you've got down to these primes. They're the indivisible numbers. So a prime number is a number you can't divide any further. And for me, that's why I like to call them the atoms of my subject. They're the numbers in which you build all other numbers. So a number like 105, I can break that as a molecule and I break it down, divide it, divide it, divide it until I get down to these indivisible numbers, which are the primes that build all the other numbers. So for example, your telephone number, if you're very lucky, you might have a prime number telephone number. I've always wanted one and I've never had one. But if it's not, then eventually, you'll keep on dividing and dividing and eventually you'll get down to these primes. And so for me, they are a little bit like the periodic table of my subject. The periodic table is the most fundamental thing really in the subject of chemistry. It's an amazing discovery by Mendeley of this kind of pattern that exists behind these atoms that you can use that pattern to predict in a very mathematical way new elements inside there. And now we have a list of basically up to 120 a few unstable towards the end. But for me, these prime numbers are really like our periodic table. But the primes, they're very, very challenging as a sequence of numbers. Because if I write the primes and I try and spot some sort of pattern inside them, there seems to be no pattern there at all. I kind of feel like there's a heartbeat of my subject. So here I've written it as a heartbeat. So every time the line runs over a prime number, it kind of beats the heartbeat of this subject. But when you look at this, it's very hard to predict when the heart's going to beat next. I mean, this is a subject which needs to visit the cardiac department. It's sort of, you know, you get sort of big gap, 23 and then a big gap, then 29, 31, these twin primes and another big gap. And actually, as you go on further and further, it seems more and more challenging to be able to predict where it's going to, when it's going to beat next. And intriguingly, mathematicians sort of feel there's much more in common between the prime numbers and these lottery ticket numbers than between the primes and these things like the Fibonacci and Triangular numbers. I mean, we don't have a formula which will tell you the hundredth prime number in a kind of an efficient way as this. And when you look at the structure of them, they have a lot of a sense of randomness in there. And we've been studying these things really for millennia. In fact, the first people to study these numbers, who do you think the first people to study prime numbers were? The Greeks, ancient Greeks, maybe? The Indians, maybe the Indians, because they were very good at math. Mesopotamians, they did amazing sort of things on astronomy with numbers and things, but actually Chinese, they explored these numbers as well, but none of those actually. There was another. No, it was actually an insect. So this insect actually started studying the primes long before we did. This insect has a very curious life cycle. It's a cicada, it lives in the forest in North America. There are some cicadas all the way around the world, but these ones are unique to North America. They have this extraordinary life cycle. They hide underground doing absolutely nothing for 17 years. And then on the 17th year, or so the same day, they emerge en masse into the forest. That's the extraordinary event. And they sing. This is the sound of one cicada. Let's see if I can make this play. You have to imagine this multiplied by 7 million. The forest is actually full of these things. And it's generally moved out of the area because it's so unbearable. And they party away. They eat the leaves. They mate. They lay eggs. And then after six weeks of parting, they all die. And the forest goes quiet again for another 17 years before the next generation appears from underground. Absolutely extraordinary life cycle. First of all, how on earth do they count 17 years? It's very rare that any energy year early or a year late. It's not quite clear. There's nothing in the natural cycle which has a 17-year cycle. I mean, 11 years, well, the sunspots maybe, but that'd be very curious if they were able to pick that up underground. But 17 is a prime number. Is that important? Is it just random that it's a prime? Well, I think there's another species of cicada in North America which has a 13-year life cycle. In fact, I got a chance to see those last year. I went out. I was making a program for the BBC called The Code. And the round national area, this 13-year cycle of cicadas, was emerging from the ground. It's absolutely extraordinary. And talking to the residents, you know, they remember when they were kids, when they last appeared, and they won't appear again for another 13 years. But around America, you find this different species, some 17, some 13. You never find 12, 14, 15, 16, or 18. So what is it about the primes which seems to be key to these cicadas? Well, we're not actually too sure, but we think there's a theory. They think that maybe there used to be a predator that also used to appear periodically in the forest. And the predator used to try and time its arrival to coincide with the cicada. Now, you can do a little thought experiment. Let's make the predator appear every six years. Okay? And the cicada. Now, we're going to have a cicada which appears every nine years. Okay. Well, if you think about that, they meet quite quickly because in year 18, the six year predator is there and the nine year cicada is there. And so they get wiped out. If you go down to an eight year cycle where they get to year 24, the first number divisible by six and eight, again wiped out. But now take it down to seven, they're appearing more often in the forest, but when do they meet for the first time? The six year predator in the seven year cicada. 32. You answer the life of the universe and everything and the survival of this cicada. So 42 years it takes before they actually get in sync. And that means the cicada has a much better chance of boarding up. Predators probably dead by then. It's gone completely hungry. But it seems like maybe there was a sort of competition in this forest where the predator sort of shifted its cycle. The cicada had to move in order to survive. And so there's a message here. You know, you matter to survive in this world, I think. So the cicada knew its primes up to 17 and the predator didn't. And so it's got left with this kind of, this prime number life cycle. So very curious. But yes, I think actually many of the civilizations that you mentioned started exploring the primes basically because they are the most basic numbers in our subject. And I will probably credit the ancient Greeks with the first really sort of analytical approach to my subject. And Euclid in particular has the sort of first, I think the first great theorem of mathematics. And it's one of the things called the elements of the greatest mathematical textbook of all time. And he said, you know, the chemists, they've got this thing, the periodic table with all the atoms in. They can get it on a nice poster. I mean, maybe mathematicians, we could just do the same thing. You'd just have a big table of all the primes. And if I need to know something about the primes, I guess this table, check it. But Euclid showed that anybody who tried to write down the primes in a big table would be writing forever. Because in one of the propositions that he proves in the elements, he proves why the primes they go on forever. You will always find another bigger prime, however far you go up. And it's a beautiful example of the power of mathematics to prove things with 100% certainty. And I think it is something unique about mathematics. If you think about the other sciences, it's the only process of evolution. It's the kind of survival of the fittest theory. And, you know, once a chink, something isn't seen to be quite right, something in physics is just an approximation you will produce relativity. It's very much a sort of evolutionary process. But mathematics is very different. In mathematics, we still teach our kids the things that were discovered 2,000 years ago by the ancient Greeks. They were true today as they were 2,000 years ago. And that's the power of proof. In all the science I think that we're teaching, you know, we're thinking about Greek chemistry, it was Earth, Wind, Fire and Water were the atoms which make up the molecular world. So I think there is something very powerful about mathematics and the power of proof to be able to get that sort of certainty about ideas and the truth. So for example, this is a beautiful proof that Euclid gave, that the primes go on forever. But despite the fact that he proved that they go on forever, he couldn't find a way to find them. It's a non-constructive proof. So, you know, if I asked you to find a billion-digit Fibonacci number, you could use the formula to find it. But the largest prime number we so far discovered has nearly 13 million digits. It's pretty big. It only takes a number with 80 digits to count all the atoms in the observable universe. If I read this number out aloud to you, it would take me two months to do it. So, you know, I'm not going to do that, don't worry. And interestingly, there's actually a project online. If you want to try and help the search for big primes, there's a little piece of software you can download, and my computer actually, whilst it's being idle here before I move the presentation on, is running this program in the background, looking for more primes out there. There are primes out there. The person who discovered, and it's just amateurs doing this, the person who discovered this prime, it was the first person to pass the 10 million digit barrier, and there was a prize of $100,000 for the first person to pass that. And I think the next, it's 100 million digit, there's $150,000 prize for that one. But I suspect, actually, probably your electricity bill will be more than that if you run this program. So, I mean, the really kind of thing is indicative of how little we understand these numbers, that although we know that they're gone forever, this is the largest one we know. We know there are infinitely many more after this, but we haven't a clue where they are. In fact, there's a million-dollar prize. It's one of our great central problems, something called the Riemann Hypothesis, which is trying to understand some sort of structure, and how are these numbers laid out? Is there genuine randomness in them? What is the sort of structure hiding behind there? And there's a million-dollar prize. It's one of the things for the Millennium Prizes. There were seven prizes offered by the Clay Institute, Landon Clay loved mathematics and wanted to sort of reward the discoveries of mathematics. But actually, I think most math... I would personally sell my soul. I'd pay a million dollars to find the solution to this, because it's such an extraordinary challenge. But as I said, they look a little bit random, but actually, even things which are random, rather surprisingly, mathematics can have something to say about it. It was one of the breakthroughs in the 17th century that randomness was considered perhaps something a mathematics could have nothing to say about. But even when things are random, we can actually use mathematics to make predictions. So I'm going to run a little experiment with you, just to show the power of mathematics to perhaps make predictions about how you behave as random individuals. One of the best examples of randomness is the lottery. So hopefully it will come in with a lottery slip. And so we're going to run a little lottery here, a continuing education lottery. I'm afraid there's no sort of million-dollar prize. It's just the honour of perhaps getting as many as possible. So could you pick six numbers? This is like the UK National Lottery. You have to pick six numbers from the numbers from 1 to 49. And then what I'm going to do is to use a little bit of mathematics to make some predictions about the numbers that you chose. Okay, so you've chosen your numbers. Okay, so I'm going to get somebody to choose some numbers here and we'll then see how close you get. So don't ring the numbers after they come out of the box. You guys should use mathematics to tell that, of course. A lot of fraud detection is about some strange patterns emerging in your behaviour. So would you do the honours of choosing the first number? So what have we got? Seventeen. Seventeen, all right. Very good choice. I play for a football team and actually I play in the number 17. This is my favourite prime number. So here we go. Next one. Thirty-five. Thirty-five. There you go. Twenty-three. Twenty-three. There's another prime number. Don't worry, they're not all primes as you've seen. Twenty-four. Twenty-four. Yeah, a tree. Okay, let's push that one on. Do this. The fifth number. Anyone getting really excited? Forty-three. Forty-three, yes. Do you assess? There's a lot of primes coming out. Here we go. Actually, forty-three is one of the twin primes. So primes can be the closest two primes can be two apart, forty-one and forty-three. So in fact, I have two twin girls and I want to try to persuade my wife that we should call them forty-one and forty-three. She wouldn't let me, so they're my private names for them. And the last one. Anybody kind of whooped with six? Number eleven. There we go. Okay, excellent. So let me do a recap of that. Eleven, seventeen, twenty-three, twenty-four, thirty-five and forty-three. Okay, now I want you all to stand up, please. I'm going to make some predictions about what you chose. So half I see the lecture, you need to get your blood running again, so you know, get those. Good. So I'm going to make some predictions about what I think you would have chosen. So I think that, I think half of you will have not got any number right at all. So please sit down if you didn't get any of those six numbers right at all. The mathematics should predict whether there's any size of students that half of the numbers that you would have chosen, well actually you look quite lucky here. Alright, okay. Right, so the next one, so there's, I think there's one in eight of you should have got two or more numbers right. So probably about 160 people here. So maybe let's sit you down if you only got one number right. And so we should have got about, I forgot to say, about 20. 20 people. So let's see, one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen. You're standing at the back because you're members of the department. So that was pretty close. Nineteen. Yeah, so the masses are doing pretty well. Okay, what about, I would predict that probably only about three of you will stay standing because it's about one in fifty chance that you get three. So sit down if you got two numbers right. Let's see how many are standing with, who got three numbers right. Okay, so there you go. So the masses are doing pretty well. Now there's four of you still standing. But I would suspect that I'll write a lot of you out with the next one. So sit down if you only got three numbers right. Yeah, so in an audience of a thousand, I might have one of you still standing. But five numbers right? Well, I need fifty five. I mean, we're planning to do the Department of Continuing Education some mass lectures with huge numbers via the internet. So if I run this experiment, I might have perhaps one person dialing in and saying they got five numbers right. And six numbers, well, you know, there is a one in a fourteen million chance of winning the national lottery each weekend. So, you know, even online, it would be great if we had one person winning. But of course, you know, that number of people play the lottery each week that's, you know, people do win because there are more than 40 million people probably playing. Now, give me a sense of the scale of these numbers because there's one thing that we're very bad at. It's assessing, you know, after a certain while, billion, trillion, you know, it's basically infinite. If I bought a lottery ticket every week, then after a year, I might have got three numbers right. We're taking 20 years buying a lottery ticket every week to get four numbers right. Now the first thing that... Okay, how many fifty... I think you have to go back to something like King Alfred, okay? If King Alfred had bought lottery tickets, you know, by now he might have got five numbers right. And to give you a sense of scale of this one, it's the first sort that Homo sapiens had with... Oh, I must have bought a lottery ticket. By now, he might have, or she might have won once and got all the six numbers right. So, that gives a sense of the scale of these things. So, okay, the math is quite good at sort of telling you things about large numbers of people and things like that. But how can math really help you? I'll show you dying for me to tell you how to use maths to win the lottery. Well, I can't produce a formula, as I said, to pull these numbers out. But what we can use mathematics for and the maths of randomness is to help anyone in place to avoid what happened in the ninth week of the national lottery. Because in the ninth week of the national lottery, something rather bizarre happened. That week, 133 people got the right answer. I mean, there was a kind of feeling like, oh, was there some sort of insider thing going on here? You know, where a lot of people cannot play. No, what happened that week? And it's an indication of actually how bad we are at randomness. We're so addicted to patterns that even when you're choosing your numbers, you've probably left some sort of patterns sitting inside there. And what we tend to do when we choose random numbers is that we tend to spread them out. We tend to spread them out and are kind of neatly across the page. So if you look at the numbers that week, they were all kind of neatly laid out. There was lucky number seven, and it was all sort of quite nice on the sort of lottery ticket. And so that's the reason. I mean, the people who won that week must have been kicking themselves. You know, they're sitting on the city and these numbers come, yes, I've got six hundred, they're going to be a millionaire. And you know, they walk away, they phone up and find they have to share their winnings. Because there were 132 people. I mean, they only won 100,000 pounds that week. And, you know, they were hoping to be a millionaire. I mean, I wouldn't say no to 100,000 pounds, but when you've got all six of the numbers right, and it's going to say, you know, another lifetime of homeless appian to go do it again, you'll get a bit frustrated. And it was because they were rather nicely laid out. And the intriguing thing is that, actually, randomness, and this is what people tend not to realize, is that randomness actually quite often clumps things together. I mean, so let me, oh, I'm going to do a test on how good you were at randomness. I want you to stand up if you've got two consecutive numbers in your lottery ticket, something like 17 and 18, or three and four. So, okay, so that's quite a few of you are standing, but actually the mathematics says that half of you, half of you should be standing because half of the possible six numbers that could come out of here have two consecutive numbers. So it's kind of an indication. This is probably, you know, it's pretty good, but it's not half, okay? You can sit down. And then as an indication, I mean, look at the numbers before the eighth week and the tenth week, the eighth week you had 21 and 22, tenth week the week after you had 30 and 31, and indeed what did we have? We had 23 and 24 coming out. So half the number of batteries I could run here will have two numbers come together. But people tend not to do that. They tend to think, oh, yeah, well, I've got 23 and 24 is not going to come out. And so actually, here's a strategy for the national lottery. If you're choosing numbers, then clock them together, because if you do win, which is highly unlikely, but if you do win, then at least you won't be sharing it with another sort of 132 people. So that is a strategy. And it goes against what the norm is that people don't really do randomness very well. How many of you chose one, two, three, four, five, six? Put your hand up if you chose one, two, three, four, five, six. Good, yes, yes, these are the true mathematicians in the room, because these people know that one, two, three, four, five, six is as likely as anything else. I mean, very unlikely, but actually, you know, if you choose that, you say, oh, that's never going to come out. Well, unfortunately, so are your numbers. They're not going to come out. But don't choose those numbers, because interestingly, 10,000 people a week in the UK choose the numbers one, two, three, four, five, six. Because I think it just goes to show actually how intelligent the UK population is that they know that that's as likely as anything else. So the mathematics of randomness is very powerful, and it's something that gets used a lot. Also, knowing when people are not being random and they're trying to be, be careful with your tax forms, because actually, there are a lot of mathematicians employed by the tax authorities who can pick up the fact that you're faking your expenses because of things like this. It's very hard to choose random numbers, and there's something called Benford's Law, which they use to actually pick up. I mean, if any of you have got six numbers right, I'd be pretty convinced that actually, there's something fishy going on. So randomness is something even mathematics can have some say at. But there's one area of sort of the natural world that mathematics, although we have a language for it, finds very difficult to navigate. And this is the mathematics of chaos theory. Chaos theory, you've probably heard quite a lot about. Chaos theory is an area of mathematics which was first discovered really by Henri Poincare, a French mathematician beginning of the 20th century he discovered this. It was actually a problem that he was working on set by the King of Sweden, which was to determine whether our solar system is stable or not. So our solar system is, you know, seems to be ticking on for many millennia, sort of in its nice regular way. But is there some point at which perhaps our solar system will kind of fly apart because of some sort of irregularity in it? And so Henri Poincare is trying to understand. So in fact, if you have two planets, sure enough, need improve that they will just, to the end of time, just carry on doing ellipses around each other. And then nothing unexpected will happen. But what Poincare discovered is that as soon as you put another planet inside it, so three planets, it's impossible to really tell. So you could have a system where the three planets are very regularly doing some sort of pattern around each other and you can prove that that will go on to the end of time. But there are other systems where the thing looks very stable. It seems to be repeating itself over and over again. But then just a very small perturbation can send the system flying off in a completely different direction. So you really don't want to be living in this planetary system because all of these planets go... And so the question was, our system, we've got a lot of planets, not just three, is it going to be stable to the end of time or is there some sort of instability? And what Poincare discovered is that actually a very small change in the initial conditions of where the planet is concurs the thing to go from stability to complete instability. And there's a lovely example of this actually. So this is the signature of chaos theory. The chaos theory might have very good equations for a system, but they may be so sensitive to very small changes that a very small change in the location of Venus could actually cause the thing to go in a completely different direction. There's a very nice example of this in terms of a pendulum. So a pendulum you think of generally is a very regular thing because a pendulum we use to keep track of time. It's so regular. But I've got a slight variation on a pendulum which is a double pendulum. So this is a bit like a leg where it's got this joint in the middle here. Let me test this thing out here. This thing, although it looks very simple, in fact you can write down the mathematical equations to this thing. It's just two rigid bodies connected. They're called to predict. And in fact, even if you run the model once, so here we go. Being able to predict, could you predict which way that's going to go around? Is it still going? Yep, there it goes again. It's absolutely extraordinary. Just such a simple thing. Two pieces of metal stuck together. But if I tried to repeat that as well, so I headed about here trying to repeat the same sequence. You have to be very careful. This is vicious actually. Let me just see whether I can... This is my favourite desktop toy. I can play with this for hours because it's so unexpected. Is it... Try to go through again. Let's do a really big one. So, you know, something so simple. Mwah! I mean... Do you know one for Christmas? I've got it in a single chaotic pendulums.com. It's a lot of fun. But, you know, that's extraordinary. You could never repeat, probably. Even if you tried, you'll have a small little bit of a difference from what you did before, and you'll get completely different behaviour each time. And there's an interesting example of that actually with a different sort of pendulum. There's another desktop toy you can actually buy, which is you have a metal pendulum and three magnets. What you do is you just set the pendulum off and it sort of whizzes around and eventually tends towards one of the magnets. But here, I ranked three computer simulations of this where I started the pendulum at almost the same place, just very small, you know, like the tenth decimal place, I just shifted the location of it. And just that small change completely changes where the pendulum ends up. The first one, it ends at the blue magnet, the second one, the red magnet, the third one, the yellow magnet. And there's another example, that even though I have the equations to describe this, if I don't know... You can never know the precise location of these things. It might be the fiftieth decimal place will still cause a different behaviour. Here's a picture describing the behaviour of this pendulum. So basically, if I start the pendulum over a red region, it means that it's going to end up at the red magnet. So, for example, here's a red magnet. Sure, if I'm just very close to a red magnet, let it go, it's just going to swing and end up there, blue, yellow. But look at these regions here. This is basically an example of something you've probably heard of called a fractal. A fractal is a kind of geometric shape which has infinite complexity. However far you zoom in, it never seems to simplify. So in this region here, basically, you know, it runs... Well, let's use this here. So you run from red to yellow to blue to yellow to blue, red to red. And just a very small change, in a completely different direction. Now, unfortunately, these kind of equations really control very many things in the natural world, and one in particular is the weather, of course. So this is why we have this thing called the butterfly effect which you may have heard about, which is the idea that the equations for this weather are actually not too complicated, but they're so sensitive to these small changes that I've been along to the Met Office and seen their models running, and what they do is that they take the data, they read the temperatures the wind speeds and things like this, and they feed that into the model and they run it, and then they make small changes in these things because you can never get complete accuracy. And when you run the model, you find five days the predictions are pretty close. But after 10 days, there's just no way you can know where it's going to end up. But they run about 50 different models, they're all saying different things. And this is why we have this thing called the butterfly effect because it might be that a butterfly flaps its wings, changes its wind velocity in one place by a very small amount, and it can cause the magnet to go from a red region, a hot day to a totally blue region, a cold day just because of that small change. So this is an important thing sometimes. Actually mathematics is a powerful tool even to tell you when you can't know things. And it's sometimes important to know you're in a region. I mean, in that computer model, if you're in the Sahara deserts, you're in this kind of region, and if you're in Antarctica, you're in this kind of region. But sometimes it's important to know when you're in a region where you can make predictions and when you're in another region where basically you have to say this is so sensitive that we cannot make predictions. That's as powerful information as well. Now this kind of mathematics affects many different sort of issues in the natural world. And one of them affects a very interesting issue about population growth. So this is a very curious question. So here's a question for you. Who do these animals throw themselves over a cliff in a mass suicide pack every four years? So any votes for the muskrats? Who think muskrats throw themselves over cliffs every four years? So one muskrat fan down here, or maybe not. I think there's another one in there. Vols? Who thinks vols? Two vols, okay. How about lemmings? Who thinks lemmings? Oh, lots of you going for lemmings. I think I'll put it on my head. Yes, so maybe you're following each other like lemmings, but yes, so the story is that, yeah, in fact it was that it is lemmings who have this kind of strange behavior that every four years they throw themselves over a cliff in a suicide pack. Now the reason this story came about was because scientists had observed that a very strange thing happened to the lemming population every four years. It just seemed to get obliterated and then it was kind of revived again, but every four years it had this sort of cycle where some of you find no lemmings around at all. And so the theory developed that perhaps they'll throw themselves over a cliff. But this thing got confirmed by an amazing piece of footage which was taken by the Walt Disney. I don't know if some of you are old enough to remember, but Walt Disney used to make nature programs. And so there's this amazing film called White Wilderness where they went out and they actually filmed. They managed to capture these lemmings. So here they are. This is a lemmings I think it's in Norway. And they managed to get footage of these lemmings coming to the edge of the cliff. And they see out there and suddenly throwing themselves off. So it was an amazing sort of bit of scientific research done here. Here they are. And then they dine. There's one here really doesn't want to go. But no, unfortunately, no, I don't want to go. And I think it eventually does go. Rather intriguingly, a few years ago, the cameraman on the shoot came clean. Those lemmings did not want to go over that cliff. In fact, what you can't see is that the crew had rigged up the spinning turntable. Somebody putting the lemmings on the turntable and so the ones instead of going over the cliff were in fact, they were on the turntable and springing off. So it was revealed that in fact there's been a whole set up on fakes. But there's still left a question about what was it that was affecting these lemmings. I mean, there certainly was evidence that they were just being obliterated every four years. Well, the intriguing thing is that we've now discovered that it was mathematics that was killing the lemmings. I'm sure most kids at school would say, oh, God, second that. But there's a mathematical formula which actually compels the population growth from one season to the next. And if you learn this mathematical formula, you start to see some sort of patterns emerging. So I'm going to do a little experiment with you actually and sort of do a slightly simplified version of this formula. So basically, it's a sort of feedback formula. So you start with a certain number of lemmings and then each season. So in the first model I'm going to run, each season the lemmings double, but there's not enough food around for everyone to survive. So this formula tells you how many lemmings will survive from one season to the next. So what you do is you take, so you start with the first generation, you double that up. Next generation, how many survive in that generation where you take the first generation multiplied by the second generation multiplied by those together, that's 10 and that's the number that won't survive. So I'm going to run this as a little experiment. I need some help. So I need some people to volunteer to be some lemmings. Good, so there's one lemming. So we've got one lemming. I'm going to start off with two lemmings. So let's just start with two lemmings. So if you'd like to come up on stage, can I have another lemming volunteer? Great, excellent. Up you come. Good. It's quite important to have a man and a woman in this as we've never got it going. Don't worry, this is mass talk. I'm not going to do any of that biology. That's kind of bad. So okay, so here are my two lemmings. The first generation, the next generation, they double up and we have two more lemmings. So I need two more lemmings to come up to be the next generation. So excellent's good. I'm going to start pointing at people anyway. So I need one more lemming. Excellent, up you come. So now, but not all of these are going to survive. So now we have to run this formula. So two lemmings doubled up to four lemmings. So we do two times four, which is eight divided by ten. So that's approximately one of these lemmings is not going to survive. So there are three will. So in order to find out who of you is going to survive, you're going to play a game of musical lemmings. So here we are, three chairs, four lemmings. Let's cue the lemming music. Okay. So which one will survive? Who are you going to survive with one lemmings? You can do a bit of lemming beading if you want. Yeah, excellent. Oh, unfortunately. So we're going to kick her off over the cliff. So there she goes. And we have three lemmings that survived. So very good. Give me a round of applause. But we haven't stopped yet. So now's your chance to be like, oh, I like to boogie like a lemming. Now's your chance. So we have three lemmings survived. But they double up now. So three, double up to six. So it's six lemmings. So I need three more lemmings. So please get involved. It's all about masses and spectators' thoughts. Excellent. I can see you've great. There's one here. Two more. How about excellent up you come? And yeah, come on. Yeah, great. So we've got six lemmings now. So as they're coming up, let's run the formula. So there's not enough food for all of them. So the formula says, OK, there's more lemmings around. So we're likely to lose more. So we do six times three as 18 divided by 10. That's roughly two. We're not going to survive. So we've got four of the wheels. I mean, you need an extra chair in here. I didn't fill out a health and safety form for this. I'm really sorry. Please don't tip over the side here. So anyway, let's cure the lemming music. We've had another two revived to dust. Oh, I like the lemming boogie in here. It's vicious. Well, I think you have been kicked off. So I'm sorry. The boogie saved her. So we'll kick these two off. There they go. And four survived. OK, so up you come. The four double up to eight. So I'm going to need another four lemmings. So now I've become great excellent. And how about you two? Yeah, yeah, yeah. And you can come as well. There you go. Unless there's something bursting to get up. So now, gosh, we've got this pretty crowded up here on the cliff face. Let's run the formula. So now we've got four lemmings. Four doubles up to eight. So four times eight, 32 divided by 10. That's roughly three are going to die. So I need another chair in there. Where are we going to stick that one? Yes, we're going to do a pentagon now. Yes, there is a geometry at work. There was a use to that geometry. OK, so be really careful over there. OK, here's the lemming music. So three is the magic number. Three are not going to survive. Five will. No hovering, madam. Unbelievable. No hovering going on. OK, let's fight it out. I think that hovering, exactly. So we'll kick them off. So three got lost. And we've got five left now. Don't worry, you're not going to have to do it again. Because something quite interesting happens now. So five of you just signed you up. So five made it through. What happens now with this formula is that five doubles up to 10. Five times 10 is 50 divided by 10, five dies. So interestingly, the population has grown, but now it stabilizes. And from now on, you just have, with this little model, five from now on. So this little model, where it doubles every season, you run the formula, you get stability. So let's give our five surviving lemons a round of applause. Thank you very much. So it happens that if I started with a lot, it would have come down to five. And so this is what happens when the population doubles. But when you change the equation, the amount of lemons that gets born each season, for example, if I tripled it, to make the lemons a little bit more productive, so it goes from, say, two to, so for example, if I ran two to double to six, then you run this formula. What you find now is that you don't get a stability, you get a ping-ponging between two values. It goes back and forth from five to eight. So that change in the production of the lemons from one season to another can produce a different behavior. So now you get this ping-ponging, and you whack it up a little bit more to 3.5. Then you start to see the behavior that you see in these lemons. So every four years, now we have four different values, which it ping-pongs between. And every fourth year, the population kind of plummets. And that's the reason that we're seeing this four-year cycle where suddenly they're none, and then it starts to pick up again. You get three years of ping-ponging around, and then it descends again. But the intriguing thing is, if you push it a little bit further and take it up, so the population quadruples every season, then suddenly this kind of rather periodic, regular behavior suddenly disappears and chaos ensues. It's like being sunny in that region, the factual region of that picture where you don't know where the pendulum is going to end up. So here is what happens when you have the quadrupling, and again, if you just throw in one extra lemming into that, it can cause the graph to go in a completely different way. So this is an important point to know, because you might think, if you look to this graph, that maybe there was some environmental factor or some human intervention, which caused the population suddenly to plummet at this stage. But using the mathematics, you can understand, no, actually this is just purely a function of the equation that runs the way the population dynamic works. So there's a lot of examples in the natural world where you get this kind of shift between chaos and regular behavior. Turbulence, for example, that's another area that we do not understand at all. Turbulence obviously affects so many things, you know, when you're flying in the airplane or a flu is flying through pipes and things, you need to know when things are going to go from this rather regular behavior to chaotic behavior. And I think that of all of the sciences, the mathematics, which is the most powerful of being able to understand the things that we can know and the things that we can't know. And if you'd like to find out more about the power of mathematics, then continuing education has actually, we've devised an online course called the Number Mysteries course. It's a 10-week course, which takes you a little bit further and deeper into these stories. You can run some experiments, join up online with other people on the course and actually find out a little bit more about the extraordinary power of this language of mathematics. So thank you very much.