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Thank you. Well, I'd also like to add my thanks for hosting us here in Solihull and to say hi to everyone out in the universe there.

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I think we, as humans spend a lot of our time thinking about the future.

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I did an exercise through the BBC. We were doing a programme about consciousness,

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and I think it's one of the unique kind of traits of consciousness that it allows us to imagine ourselves into the future.

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I have a cat at home and my cat just sits there enjoying the day.

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I don't think it spends very much time thinking about what's going to happen next,

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but it's actually one of the best tools we've developed to enable us to try and look into the future and make predictions.

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I think his on subject of mathematics actually is very curious.

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When I'm at a party and somebody asks me, What do you do? And I say, I'm a mathematician.

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I don't know whether you find this alone,

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but with this kind of look of horror appears on the person's face and they back away sort of their glass becomes very empty,

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very quickly and I before they go. They always seem to tell me what they got in the sea level.

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I'm not sure what that's about, but anyway,

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I'm very persistent and I chased after them and I began to realise that mathematicians were very misunderstood breed.

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So I think most people think now, what are we doing in our offices in Oxford all day as mathematicians and as I've talked to people,

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I've got this feeling that they think I'm doing long division to a lot of decimal places.

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And, you know, surely my computer has put me out of a job by now.

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But I tried to explain to them that no, actually mathematics.

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If I was going to define mathematics, I would define it as the science of pattern searching.

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And it's not ability to understand a pattern which gives us this great tool to be able to look into the future.

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Because by looking at a pattern of data in the past, if we can find that pattern,

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then we might be able to read that pattern so we can predict what it's going to do in the future.

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How do we know about so what's the trajectory of our universities?

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Because we've seen the data of the universe from the telescopes that we've seen, it's amazing.

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We're stuck on this planet. Yet through the information we've got,

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we're able to see that we can work our way backwards to see we came from a Big Bang and voids to see what will happen to the Universe in the future,

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even on our smaller, more local scale of the universe by looking at the data of the climate.

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We're beginning to understand what might be happening in the future.

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We're going to have to use our science and mathematics to actually do something about what that trajectory looks like.

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So I think for me, mathematics is all about looking for patterns.

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So what I would love to do in this presentation is to show you some of the power of mathematics to predict things,

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but also some of the limitations that mathematics can't let us know everything about the future.

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I think that time the challenge of patent searching for me is kind of perfectly encapsulated in the

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challenge you probably had at school where you're given a sequence of numbers and you have to try and spot,

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well, what's the logic behind this sequence of numbers to be able to tell what's going to happen next?

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So I thought I'd see how good you are here in Solihull at predicting what's going to happen next.

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So I've got a series of challenges for you, which gets a little bit more difficult as we go on.

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So a sequence of numbers one I want you to do is can you work out what the pattern is and be able to predict what's going to happen next?

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So we'll start off with a very easy one one three six, 10, 15, 20, one 20 and what's the next number in that sequence?

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Thirty six. Very easy. So how did you get thirty six?

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Yeah, you know what I'm pointing you use see, we're very loud and between the first two is two and three four.

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Exactly. You're sort of adding one more number on each time one plus two plus three plus four.

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In fact, we call these the triangular numbers because you can view them very geometrically its number of stones.

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You need to make a triangle. How many stones you need to when you add on an extra layer each time.

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And actually, if you wanted to know, for example, what the hundredth triangular number is,

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so how would you know way into the future what this number sequence is going to do?

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Well, you could do the kind of the dumb way would to be add up all the numbers from one to 100.

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But mathematicians, we're not very good at kind of arithmetic, actually. So I'd probably make a mistake if I did that.

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But what we want is kind of to understand maybe a formula which will tell us a smart way to understand what the hundredth number might be.

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And here's a trick I often use in my own research, which is kind of changing the language.

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So instead of considering numbers to get this formula, if you consider these things more geometrically, here's some triangles.

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If I put two triangles together, I get a rectangle.

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Well, counting things in a rectangle is easy because it's the the base times the height, so I can calculate the number of stones there, a triangle.

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I just have that that number, and then I have a formula which will tell me the hundreds and thousands any number in this sequence.

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So this is a sequence we understand very well.

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OK, well, if you're if you read the da Vinci Code, you're not allowed to play on this one because this is the one of the challenges.

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First challenges Dan Brown sets you in the da Vinci Code. So what's the number in this sequence?

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One one two three five eight 13 21. Thirty four.

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OK, a very famous sequence. You got that by adding the two previous numbers together.

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So these are called the Fibonacci numbers named after this Italian mathematician in the 12th 13th century,

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who understood that these are nature's favourite numbers. You find them all over the natural world.

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So for example, if you count a fruit, open an apple sort of horizontally, you find a five pointed star inside or a banana.

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You'll find a three pointed star, a Sharon fruit,

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an eight pointed star pineapple means you'll find these numbers spiralling up and down flowers if you count the number of petals on a flower.

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Invariably, it's a number in the Fibonacci sequence. And if it isn't the number in the Fibonacci sequence,

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that's because a petal has fallen off your flower, which is how mathematicians get round exceptions.

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And so actually, these numbers shouldn't be called the Fibonacci numbers.

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I discovered quite recently that they were discovered before Fibonacci ever found them in nature, not by mathematicians,

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but by Indian musicians and poets who understood that these numbers also help you to count the number

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of rhythms that are possible if you're a tabla player that you can make out of long and short beats.

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So the wonderful thing is you find mathematics all over the place, not just nature, but also in the world of art as well.

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Now this sequence, we learnt quite a lot about this sequence.

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We have a formula again which will calculate the hundredth Fibonacci number without having to add all the pairs all the way up to one hundreds.

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It's a bit more complicated than the triangular numbers involve this one phone number, the the golden ratio.

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You have to take powers of the golden ratio, but there's a lot of mysteries we still don't know about these numbers.

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So there's still some things no mysteries to solve about the Fibonacci numbers.

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OK, the next sequence some one two four eight 16.

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Thirty two is a possible answer, but thirty one is also an equally valid answer to the sequence.

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You mean it's OK, this is just doubling.

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Thirty two seems to be the most obvious, but why if any one shot out thirty one, would that be equally correct?

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Well, these numbers up to 16, are actually counting something called the circle division numbers.

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So what are these? If only take a circle. And I took one point in that circle.

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I just got one region, the circle in the middle. But now let's pick another point and join these two points up.

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Now I've got two regions, so let's keep on going well and another points.

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Join up all the lines. I've got a triangle in the middle, three regions around the outside, that's four regions.

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OK, so you think, OK, well, I'm starting to spot the pattern here at a point.

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Another one. We join all of the four points up. Well, this nice little envelope in the middle for triangles there.

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Four regions round the outside. Eight you keep going.

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What about five? Sure enough, you get this lovely five pointed star Pentagon 16.

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And by now you think, Oh, it's pretty obvious. Every time you add a points, you somehow double up the number of regions.

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And so you make a conjecture that the pattern is going to go in this direction, just double.

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And then you get a surprise because when you put six dots on the circle and join these up, the largest number of regions you can get is thirty one.

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So for me, this is quite one of the assignments of mathematics because they're often kind of surprises out there.

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Unexpected directions that something can go in. So there is a formula for the number of regions.

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It's not tumbling. It's a slightly more complicated formula.

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It's this cortical polynomial way you raise the number of points on the circle to the power four and do a further calculation.

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But this formula tells us how this number of regions grows and grows much slower than just doubling.

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And this might be a practical problem where you're actually got some sort of

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network and you need to know how many regions this can be divided up into.

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So you're going to need to know how this pattern emerges. And then you won't have quite as many as you think if it's just simply doubling.

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So another warning here you see the time a pattern might be starting to grow, but suddenly the thing can go off in a completely new direction.

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So you have to be prepared for surprises in mathematics, which is one of the reasons it's so much fun in a way.

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OK, so oh, and then a little bit more. That's that spoils my OK.

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So if anybody saw quite what popped out there, a little bit more challenging sequence two nine, 10 11 13 16.

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Usually, this has the mathematicians really working hard because they see some sequence going here,

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but as as I fleshed out, probably too quickly there.

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These don't have any pattern at all because these are in fact, the National Lottery winning numbers for the 28th of September.

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So another warning here that not everything has a pattern to it.

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So you have to pick your battles carefully when you're looking at data, and quite often there can be things that you think of patterns,

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but which actually just the result of kind of randomness in the pattern can kind of just disappear.

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But in a little bit, I'll show you that even when something is as random as the National Lottery,

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mathematics has tools to be able to make quite powerful predictions about even things which are genuinely random.

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OK, what about the last sequence? Two three five seven 11 13 17 19?

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Twenty three, exactly, and these are the the prime numbers.

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So these are probably the most important numbers in the whole of mathematics,

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the invisible numbers, the numbers which can't be divided by anything except themselves.

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And one and I would say of all of the numbers in mathematics,

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these are the most important because these are literally the building blocks of the whole of our subject.

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One take a number like one hundred and forty five. Is that a prime number now clearly divisible by five?

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So let's divide by five. Go down to twenty one times. Five is twenty one a prime number now because that's divisible by three and seven.

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But now you see our divided and divided, and now I've got down to three numbers which are indivisible.

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These can't be divided any further. And these are the building blocks which make the number one hundred and five.

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So this is true of any number. You take your telephone number, for example, it might already be a prime number,

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in which case I'm incredibly envious because I've always wanted a prime number telephone number.

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I've got an even telephone number. It's completely rubbish. But so I will swap with you if you have a prime number.

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But if not, here's a challenge for you.

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Go back and actually keep on dividing it until you get down to the primes which built which build your telephone number.

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And so every number is built out of these primes.

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For me from 30th magicians, these numbers are really like the hydrogen and oxygen of the world of mathematics.

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Probably the most important thing in chemistry. Anybody who's had chemistry is today at school.

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They will have had the periodic table up on the wall because that is the ingredients from which our universe is made.

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And Mendel have spotted a pattern in these atoms, which enabled him to make predictions about atoms that were missing from this list.

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So that's the power of spotting patterns.

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In this case, mentalist kind of rule of eight allowed him to to lay out the things and see the gaps and make predictions about things.

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It wouldn't be that so. So the chemists have kind of found their pattern, but for the mathematicians, we haven't found our pattern.

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These numbers, we know, go on forever. There are infinitely many of them.

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However, this is the largest prime number we've found so far.

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I mean, it's pretty big, too, to the panel of seventy seven million two hundred thirty two thousand nine hundred seventeen.

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Well, that's very divisible. We take one off. That's no. You get a prime number.

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This has over twenty three million digits. Bernie Bink number.

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If I said all the digits out loud, I think I calculated that we'd be here for about three months.

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So don't worry, I'm not going to do that. But this is kind of shows that we really don't understand these numbers because we've only

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asked you for a billion digit triangular number or a billion digit Fibonacci number.

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Then we could use the formulas that we found to be able to generate those quite quickly.

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But prime numbers, we don't understand a formula to generate primes.

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We don't understand a pattern which will help us to to find them quickly.

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So although there are infinitely many of these numbers, they still represent one of the greatest mysteries.

148
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We're kind of still waiting for the minda laff of mathematics to be able to to give us an understanding of how these numbers work.

149
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So it's one of the things that I work on in my research in Oxford is trying

150
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to understand these numbers because they are so fundamental to our subject.

151
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So hopefully,

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maybe one of you out there will make your name in mathematics by telling us how these numbers work in France if you want to get involved already.

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If you want to find perhaps the next biggest prime number, this was actually not discovered by a professional mathematician,

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but by an amateur who downloaded a piece of software onto his computer.

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And the software uses a little algorithm that we've discovered, which helps to find these special primes called Amazon Prime's.

156
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And there's actually prises, I think, $3000 for the next biggest prise that you discover if you pass the 100 million digit mark.

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There's a hundred and fifty thousand dollar prise for that and a billion digits, so you can get one is bigger than that.

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We know it exists because of the proof for the ancient Greeks. There's a two hundred thousand dollar prise for that one.

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Well, I think probably the electricity bill you'll run up with your computer running this algorithm may be greater than two hundred thousand dollars.

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But if you can understand the true mystery of these numbers, how they are laid out in the universe of numbers,

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in fact, that's one of the things that we call one of our millennium prises.

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There is a million dollar prise offered by a businessman in America.

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For anybody, you can truly understand the mystery of how the primes are laid out.

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So, so these are really one of our biggest no mysteries in the whole of mathematics.

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In fact, we think there may not be any pattern at all, which is sort of harder to prove than there is a pattern.

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And we think that the primes the way they're laid out has much.

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More in common with these lottery ticket numbers in the triangle or Fibonacci numbers,

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and that's somehow one of the challenges of this problem called the Riemann hypothesis, which is to show why there isn't really a pattern.

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I mean, how do you show there's no pattern that's much more difficult than actually finding a pattern to help predict them?

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But even when something doesn't have a pattern, when something's genuinely random,

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still, you're able to tell some interesting things about these numbers.

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So, for example, let's take the National Lottery. In fact,

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mathematical competitions are employing it to make a lot of predictions about what we expect will happen each week when we run the National Lottery.

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So I thought what we would do to explore the sort of power of mathematics to make predictions is we're going to run a little lottery with you.

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So you've all been given a lottery form as you came in and just seen it in the National Lottery, you have to choose six numbers.

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So if you've got a pen or borrow a pen from a neighbour, perhaps circle your numbers, your six choices.

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And then I got a box of numbers here and not quite as grand as the one in the National Lottery will try to make it look glamorous with my gold paper.

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So a little sorry. There's no prises.

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I'm afraid I'm not going to give you a million dollar prise. So just the glory of getting as many as you can.

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Please choose your numbers now and not when they come out of the box.

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I know it's very tempting to say just have a little look over with your neighbour has actually done it.

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And that's another thing. Actually,

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mathematics is very good at working out when somebody has cheated because the chances that somebody gets all six of these numbers right is very,

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very rare, as you will see. And probably if one of you does.

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I would predict probably you, you were waiting for the ball to come out.

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OK, so you've already chosen your numbers. You can play this actually if you're watching my Facebook Live.

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Just write down some numbers six numbers between one and 49.

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And let's see. Maybe actually if we've got thousands are watching online.

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Somebody might actually get six, but they can tweet us live and see what they get.

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OK, so are you ready? Let's get some numbers to be chosen. So if you could choose our first number, please?

191
00:18:30,950 --> 00:18:36,200
Great number seven grades, a prime number of rights, yes, think of that number seven.

192
00:18:36,200 --> 00:18:43,010
I bet you've only got seven because that's magicians use that always because people always choose seven, because it's their lucky number.

193
00:18:43,010 --> 00:18:49,420
Number three, you are, yeah, good on primes already. Yeah. Number 10, OK.

194
00:18:49,420 --> 00:18:55,040
Online, we get 10. Number of digits we use on our hands.

195
00:18:55,040 --> 00:19:01,220
Thirty one. Yes. Quite prime laden number one so far.

196
00:19:01,220 --> 00:19:08,200
Are you going on a great, excellent number for a lot of low lying numbers here?

197
00:19:08,200 --> 00:19:16,780
And finally, number thirty three. OK, so I didn't hear any Woohp say probably, no, but you got all six, right?

198
00:19:16,780 --> 00:19:21,640
So let me put these in order. Oops. Come back. That's one, right?

199
00:19:21,640 --> 00:19:29,980
Just put these. So we've got about three four.

200
00:19:29,980 --> 00:19:35,780
It's three four. All right.

201
00:19:35,780 --> 00:19:39,670
Seven, 10. OK, so I'll just read them out again.

202
00:19:39,670 --> 00:19:48,460
So we've got three. Number three, number four, number seven, number 10, number 31 and number thirty three.

203
00:19:48,460 --> 00:19:52,480
Great. Now about these points in a lecture.

204
00:19:52,480 --> 00:19:55,490
I always get a little bit sleepy and I'm starting to drift off.

205
00:19:55,490 --> 00:20:02,740
So what I'm going to get you to do is to all stand up, please, OK, and just get a little bit of blood running through your body.

206
00:20:02,740 --> 00:20:07,960
That's great. Yes, I want to wave your things. Let's see all these lottery papers.

207
00:20:07,960 --> 00:20:12,670
Great. I'm going to take a picture of you all say, Yeah, that's fantastic. Excellent.

208
00:20:12,670 --> 00:20:18,040
I shall tweet that later. Good. Right. Keep standing. Now I'm going to make a few predictions about the numbers,

209
00:20:18,040 --> 00:20:26,140
but I think that you chose because I'm going to make a prediction that I think half of you didn't get any numbers right at all.

210
00:20:26,140 --> 00:20:30,190
So I want you to sit down. If you didn't get any numbers right at all.

211
00:20:30,190 --> 00:20:34,070
And let's see whether how good the maths is. I'm making a prediction.

212
00:20:34,070 --> 00:20:37,930
So I think that's I think you're a little over the half.

213
00:20:37,930 --> 00:20:45,670
And maybe that's an indication that we got a few lower numbers in there, which is probably in some of your choosing birthdays, maybe or something.

214
00:20:45,670 --> 00:20:55,210
OK, but it's still pretty good. So let's see what about. So if I was going to ask you to sit down if you only got one number right?

215
00:20:55,210 --> 00:21:00,070
Let's see. We should. So this is an eighth of you.

216
00:21:00,070 --> 00:21:04,690
So let's see. We've got about it's not about an eight standing.

217
00:21:04,690 --> 00:21:09,040
I would say. That's pretty good. Yeah. So but what about three numbers?

218
00:21:09,040 --> 00:21:13,840
OK, so one in 50 chance. So I would say we've got about 200. So maybe five of you will standing.

219
00:21:13,840 --> 00:21:18,340
Let's make a prediction. Five of you would got three numbers or more, right?

220
00:21:18,340 --> 00:21:25,660
So keep saying you've got three or more numbers, right? This would just just her.

221
00:21:25,660 --> 00:21:30,340
Oh my God, you're so unlucky and so on the whole. Oh, well, that's OK.

222
00:21:30,340 --> 00:21:34,780
So of course, these predictions are only in the corner. You're sitting down already.

223
00:21:34,780 --> 00:21:37,480
You didn't get four, right? You only got three, right?

224
00:21:37,480 --> 00:21:42,850
OK, so four would be signing to push you to not be paid a little bit more suspicious about your mum there about is your mum.

225
00:21:42,850 --> 00:21:52,380
And she was actually because four we need one in a thousand will be getting four right one in fifty five thousand eleven.

226
00:21:52,380 --> 00:21:55,000
What he's maybe going fifty five thousand people watching on Facebook.

227
00:21:55,000 --> 00:22:00,310
Certainly some of our lectures that we live from Oxford are getting into the sixty thousand range.

228
00:22:00,310 --> 00:22:06,160
We might have one of the people watching who's got five numbers right, but six numbers.

229
00:22:06,160 --> 00:22:11,230
We haven't quite hit this many audience watching online as 14 million,

230
00:22:11,230 --> 00:22:18,580
but it's a one in a 14 million chance that you would get all of those six numbers right?

231
00:22:18,580 --> 00:22:28,390
Give you a sense of the scale of these things. So I think one of the things that as humans, we are very bad at is assessing large numbers.

232
00:22:28,390 --> 00:22:33,490
So one in 40 million is almost like, well, that's never going to happen.

233
00:22:33,490 --> 00:22:40,980
And but of course, one in a 40 million well, there are 14 million people probably playing the lottery each week, which is why one person wins.

234
00:22:40,980 --> 00:22:49,180
So space, you bought a lottery ticket every week. How long would it take you to get four five six numbers, right?

235
00:22:49,180 --> 00:22:53,290
Well, OK, you bought a lottery ticket every week after one year,

236
00:22:53,290 --> 00:23:01,150
you might have three numbers one in fifty seven, but it took 20 years to get four numbers right.

237
00:23:01,150 --> 00:23:07,180
So I worked out to get five numbers right. If King Alfred's have been buying lottery tickets.

238
00:23:07,180 --> 00:23:16,210
He might have got five numbers right. But to get all six numbers right, I think the first Homo sapiens that ever involved,

239
00:23:16,210 --> 00:23:23,440
if the first thought he or she had was I must go buy lottery tickets and went down and started buying lottery tickets every week.

240
00:23:23,440 --> 00:23:27,460
Then by now, they might have once got all six numbers, right?

241
00:23:27,460 --> 00:23:31,810
So this gives you a sense of the scale of these large numbers.

242
00:23:31,810 --> 00:23:40,060
So although you know, I'm I'm concerned I've got a fairy here to help you to predict the numbers that will come out of here.

243
00:23:40,060 --> 00:23:46,420
But the mathematics can help you to kind of work out what the chances of getting these numbers.

244
00:23:46,420 --> 00:23:53,710
But what mathematics can help you to do is to avoid something rather strange that happened in the ninth

245
00:23:53,710 --> 00:24:01,360
week that the National Lottery was run in the UK because in the ninth week of the National Lottery.

246
00:24:01,360 --> 00:24:06,130
An amazing hundred and thirty three people won the jackpot that week.

247
00:24:06,130 --> 00:24:12,490
And I mean, imagine you're at home, you're sitting on the city and the numbers coming out and you go, My gosh, I've got five.

248
00:24:12,490 --> 00:24:18,190
I'm go to all six numbers, right? You phone up thinking you've become a millionaire and then you discover you're going to

249
00:24:18,190 --> 00:24:22,120
share the million with another one hundred and thirty two people who've won that week.

250
00:24:22,120 --> 00:24:27,640
This is the worst week to win the National Lottery. So, so what happens that week?

251
00:24:27,640 --> 00:24:33,820
So I mean, there's so many people we usually use two or three, one, maybe none, but one hundred and thirty three people.

252
00:24:33,820 --> 00:24:42,220
I mean, was it somebody cheating? Was it somebody the National Lottery who ran it the film in advance and that people knew what the numbers were?

253
00:24:42,220 --> 00:24:49,390
What happened that week is evidence of the fact that as humans, we're very bad at behaving randomly.

254
00:24:49,390 --> 00:24:55,330
We tend to leave patterns wherever we go. In fact, we see a pattern rich.

255
00:24:55,330 --> 00:25:00,430
This is one of the reasons that companies that have access to our data.

256
00:25:00,430 --> 00:25:09,490
People like Facebook and Google are able to use the data so powerfully to make predictions about what we're likely to do next.

257
00:25:09,490 --> 00:25:17,230
So the numbers that we leave just tell so much about us and the numbers that you've chosen probably tell a lot about you as well.

258
00:25:17,230 --> 00:25:22,330
You see, what happened that week is that very often people, when they choose numbers randomly,

259
00:25:22,330 --> 00:25:26,830
they like to space the numbers out rather evenly across the page.

260
00:25:26,830 --> 00:25:35,440
And this is very common. And if you see the numbers that came out of the lottery that week, they were nicely, evenly spaced.

261
00:25:35,440 --> 00:25:41,980
So it means that it favoured all of those people, like two evenly spaced their numbers, which is the majority of humans.

262
00:25:41,980 --> 00:25:47,860
And so when it is nicely spaced like this, it means that a lot of people are likely to win the jackpot.

263
00:25:47,860 --> 00:25:52,900
So let's I'm going to ask you a question now about how randomly you behaved.

264
00:25:52,900 --> 00:25:59,020
So I want you to stand up if you choose two consecutive numbers.

265
00:25:59,020 --> 00:26:07,300
So seventeen eighteen, twenty one twenty two. So let's see, because those obviously aren't very evenly spaced out.

266
00:26:07,300 --> 00:26:11,050
OK, so what numbers did you choose it? Six and seven.

267
00:26:11,050 --> 00:26:18,970
OK, there you go. So now quite a lot of you are standing, but if you were genuinely behaving randomly, half of you,

268
00:26:18,970 --> 00:26:25,540
half of you should be standing now because half of the numbers that come out of this box will have two consecutive numbers.

269
00:26:25,540 --> 00:26:31,810
In fact, tonight we had three and four come out of there the previous week to the National Lottery,

270
00:26:31,810 --> 00:26:36,220
the eighth week we had twenty one in 20 to 30 and 31 came out the next week.

271
00:26:36,220 --> 00:26:40,570
So if you were behaving randomly, half of you should be standing.

272
00:26:40,570 --> 00:26:46,060
Now I want you to sit down unless you chose the six numbers one two three four five six.

273
00:26:46,060 --> 00:26:51,080
Did anyone choose one two three four five six? Let's see if anybody is standing.

274
00:26:51,080 --> 00:26:57,430
What would you what? What does this mean? Why the date? What you dayton's 40, 44 45 46.

275
00:26:57,430 --> 00:27:02,080
Oh OK, I can see what you're saying there. Yeah, right? Yeah, that's in the spirit of what I'm asking.

276
00:27:02,080 --> 00:27:06,130
So that's fair enough. So would you choose one two three four five six? So let me.

277
00:27:06,130 --> 00:27:08,710
These are the genuine mathematicians in the room,

278
00:27:08,710 --> 00:27:16,840
because these are the ones that realise that one two three four five six is as likely as anything else to come out of this box, namely very unlikely.

279
00:27:16,840 --> 00:27:20,660
So you're probably saying, well, why don't you just one two three little five six? That's never going to come out?

280
00:27:20,660 --> 00:27:24,700
Actually, we did not quite go quite close there. OK, good.

281
00:27:24,700 --> 00:27:29,410
But you do choose one two three. So you can sit down. Thank you. Five.

282
00:27:29,410 --> 00:27:34,990
If you don't choose that because apparently ten thousand people a week to that thing just goes to

283
00:27:34,990 --> 00:27:40,060
show how intelligent the UK population is because they realise that's as likely as anything else.

284
00:27:40,060 --> 00:27:42,100
So actually, what you need to do is what this guy here did,

285
00:27:42,100 --> 00:27:47,740
because that's very unusual to choose the six at the end rather than the six at the beginning.

286
00:27:47,740 --> 00:27:54,940
So. So this is a way to help you in the National Lottery, which is OK.

287
00:27:54,940 --> 00:28:00,730
You're not going to be able to have a formula to predict these numbers, but when you win, you want to take away as much money as possible.

288
00:28:00,730 --> 00:28:07,390
So clump your numbers together. And what you want to do in any investment basically is to go against the tide.

289
00:28:07,390 --> 00:28:14,740
Such a win, you win, you win big. But mathematics can also help you to work out when something looks really

290
00:28:14,740 --> 00:28:19,090
bizarre and strange is actually something you would expect out of randomness.

291
00:28:19,090 --> 00:28:23,110
So we've seen that clumping together is something you expect now.

292
00:28:23,110 --> 00:28:28,690
Something really weird happened in the Bulgarian lottery on in 2009,

293
00:28:28,690 --> 00:28:35,890
and these are the winning lottery ticket numbers on the 6th of September four fifteen twenty three twenty four two consecutive numbers came out.

294
00:28:35,890 --> 00:28:40,720
Thirty five and forty two satellites, the universe and everything.

295
00:28:40,720 --> 00:28:49,180
Nothing unusual there. Except that the following four days later, on the 10th of September, the same six numbers came out.

296
00:28:49,180 --> 00:28:54,250
Now that looks really suspicious, especially when it's Bulgaria.

297
00:28:54,250 --> 00:28:59,200
I mean, No. Three, Sorry. Anyone watching from Bulgaria, I'm not causing any aspersions on my.

298
00:28:59,200 --> 00:29:03,350
So I mean, they're bringing in the. Gary in government said, Oh, come on.

299
00:29:03,350 --> 00:29:08,780
Something is really going on here. There's something, you know, clearly some corruption going on.

300
00:29:08,780 --> 00:29:17,600
They've run the the video twice or whatever. So they started in an investigation and I couldn't find anything weird going on.

301
00:29:17,600 --> 00:29:21,050
Mathematicians got involved in this and they started to do some calculations.

302
00:29:21,050 --> 00:29:29,060
We've had a lot of lotteries running across the world every week in different territories, different countries, different counties.

303
00:29:29,060 --> 00:29:36,500
By now, we realise that this clumping means that this should happen at some stage,

304
00:29:36,500 --> 00:29:41,720
at some stage where it should be a lottery, where six numbers come out consecutively.

305
00:29:41,720 --> 00:29:50,300
So we now know that although this looks very weird, it's something that you should expect from the random behaviour of numbers coming out of this box.

306
00:29:50,300 --> 00:29:57,940
So it's a useful tool. Mathematics can show you when something looks like it might be special that it isn't so special, after all.

307
00:29:57,940 --> 00:30:03,410
My wife, you, for example, she thinks she's psychic because, you know, she'll think about somebody and they'll phone.

308
00:30:03,410 --> 00:30:09,440
But she doesn't remember all the times that she thought about somebody and they didn't phone, and that one is the one that speaks.

309
00:30:09,440 --> 00:30:15,500
But I mean, she might well be psychic donate. So that's a lot about the mathematics of randomness.

310
00:30:15,500 --> 00:30:21,440
But now I want to come to an area where mathematics is really having problems about making predictions.

311
00:30:21,440 --> 00:30:30,500
And this is really important because it's an area of mathematics and science which impacts on many things we would like to know about our universe.

312
00:30:30,500 --> 00:30:41,690
And this is the mathematics of chaos. Chaos is a mathematical idea of chaos was first discovered by this French mathematician, Andre Grey,

313
00:30:41,690 --> 00:30:50,900
who is actually trying to solve a problem that was set by the King of Sweden as a challenge for his birthday.

314
00:30:50,900 --> 00:30:58,340
Actually, the king of Sweden so loved maths, that's the thing he wanted for his birthday was to set the world at some mathematical challenges.

315
00:30:58,340 --> 00:31:05,930
And the challenge that you sent mathematicians for that birthday was a question about our Solar System.

316
00:31:05,930 --> 00:31:14,120
He was interested to know whether our Solar System is genuinely stable or whether there might be chance that at some point in the future,

317
00:31:14,120 --> 00:31:20,840
our Solar System might fly apart there. All the planets were just kind of do something really strange.

318
00:31:20,840 --> 00:31:26,870
Now it looks at the Solar System is pretty predictable because we're really good at

319
00:31:26,870 --> 00:31:31,910
making predictions about when there's going to be a transit of Venus or or an eclipse,

320
00:31:31,910 --> 00:31:40,730
or we're able to know quite a lot about the comets that are coming in and out of our trajectory on the Solar System.

321
00:31:40,730 --> 00:31:47,090
So it looks like our Solar System is stable, but can we mathematically prove that's true?

322
00:31:47,090 --> 00:31:53,270
So one can say, OK, I'm going to take our mathematical tools. It's just a bit of Newtonian physics gravity.

323
00:31:53,270 --> 00:32:03,110
We should be able to crack this one. Now, Newton had actually already shown you that if you have two bodies, say Sun and a planet,

324
00:32:03,110 --> 00:32:07,010
that they will just carry on doing ellipses around each other till the end of time.

325
00:32:07,010 --> 00:32:13,280
So these are very stable. There's nothing that's going to disrupt this. Once it starts, it will just keep on repeating that pattern.

326
00:32:13,280 --> 00:32:21,140
But when you throw in another planet into this little solar system, very strange things start to happen.

327
00:32:21,140 --> 00:32:27,470
So you can set up a very stable system where these three planets are now doing ellipses around us.

328
00:32:27,470 --> 00:32:29,360
But these with three planets,

329
00:32:29,360 --> 00:32:38,030
a very small change in the initial conditions of one of those planets can cause something that looks stable to suddenly fly apart.

330
00:32:38,030 --> 00:32:41,630
So here these three planets seem to be starting in a similar way,

331
00:32:41,630 --> 00:32:47,940
but actually on May one of this planet slightly smaller than the other ones and suddenly, after a while, these planets just fly apart.

332
00:32:47,940 --> 00:32:50,900
I mean, you really don't want to be living in this solar system.

333
00:32:50,900 --> 00:32:57,740
So but it was just a very small change in in the system caused the thing to behave in a completely different way.

334
00:32:57,740 --> 00:33:07,070
And this is the signature of chaos because I think many, many people believed that, you know, Newton,

335
00:33:07,070 --> 00:33:14,930
to come up with the laws of nature that the laws of motion and we got the equations for how the things evolve,

336
00:33:14,930 --> 00:33:20,960
that, you know, if we know the situation now, we run these equations, we'd be able to predict what's going to happen in the future.

337
00:33:20,960 --> 00:33:24,830
And OK, we may not have a precise description of the universe now,

338
00:33:24,830 --> 00:33:29,570
but a good approximation should at least give us a good approximation of the future.

339
00:33:29,570 --> 00:33:37,010
But what's point understood is that even a very small error, as we've seen here in our description of the present,

340
00:33:37,010 --> 00:33:44,540
can cause two different models to go in completely different directions and predict completely different answers.

341
00:33:44,540 --> 00:33:50,390
I think my favourite example of chaos is actually a very simple system, so it doesn't, you know, this is quite simple.

342
00:33:50,390 --> 00:33:58,400
It's just planets are basically just like little points. The mathematics is not complicated and it doesn't appear complicated.

343
00:33:58,400 --> 00:34:06,890
And this is again a very simple system. This. Is a pendulum now pendulum is Galileo understood, are so predictable, we use them to keep track of time.

344
00:34:06,890 --> 00:34:16,070
But this is a slightly different pendulum. This is I've got two pieces of metal jointed together here, a little bit like a leg.

345
00:34:16,070 --> 00:34:22,190
And so the geometry of this is totally straightforward is to rectangle similar lengths.

346
00:34:22,190 --> 00:34:25,280
The physics is fairly straightforward. It's just gravity.

347
00:34:25,280 --> 00:34:29,840
This is made to be practically frictionless, so it doesn't really contribute anything to this.

348
00:34:29,840 --> 00:34:38,680
But trying to predict what this thing is going to do next is almost impossible.

349
00:34:38,680 --> 00:34:44,960
Wow. Gone, gone.

350
00:34:44,960 --> 00:34:48,140
Oh, yes.

351
00:34:48,140 --> 00:34:55,460
So I think the fact that you're laughing is already an indication of that unpredictability because I mean, what is laughter is a response to the.

352
00:34:55,460 --> 00:34:57,920
I wasn't expecting it to do that.

353
00:34:57,920 --> 00:35:05,150
But the real challenge of chaos is it's not just we'll have a sliding weird behaviour is the challenge of repeating what we just did.

354
00:35:05,150 --> 00:35:10,070
So I've got a little notch here, which I try and predict, try and repeat exactly the same behaviour.

355
00:35:10,070 --> 00:35:13,670
So you remember what it just did? OK, well, maybe not spare time.

356
00:35:13,670 --> 00:35:23,060
Let's see whether we can get the same thing going. So to begin with these on, so fairly similarly, but some very quickly, what it's doing is so.

357
00:35:23,060 --> 00:35:27,410
Well, you see, there we go. It sort of. So it's got a little tired now.

358
00:35:27,410 --> 00:35:31,020
I've got time quickly. So we've got a completely different behaviour after a while.

359
00:35:31,020 --> 00:35:37,100
So that's the signature of chaos. This small change can cause a completely different behaviour.

360
00:35:37,100 --> 00:35:40,310
This is my favourite desktop toy. I can play with this for absolutely hours.

361
00:35:40,310 --> 00:35:47,410
So if you'll indulge me, let's do one more of those things they offer you.

362
00:35:47,410 --> 00:35:53,610
Oh, oh, it's really showing off now, gosh, well, look, know that.

363
00:35:53,610 --> 00:35:57,870
We are dancing back and forth of hips. Yeah, good go round.

364
00:35:57,870 --> 00:36:02,280
You got any more. Now he's shaking his head, OK, I guess.

365
00:36:02,280 --> 00:36:06,270
So that's actually one of my other favourite toys.

366
00:36:06,270 --> 00:36:11,730
Is this one? I've got this on my desk at home. So this is again a pendulum.

367
00:36:11,730 --> 00:36:15,960
And there are six magnets which attract the pendulum and overreach magnets.

368
00:36:15,960 --> 00:36:20,820
It's an answer to a question. So if, for example, you can ask a question, it will answer yes.

369
00:36:20,820 --> 00:36:27,720
Maybe. Definitely no way. Try again. Ask a friend. So I use this to make all my decisions in life.

370
00:36:27,720 --> 00:36:33,930
So. So I'm a big Arsenal fan. So is there a chance we're going to win the Europa League this year?

371
00:36:33,930 --> 00:36:38,610
That's probably all we're possibly going to do. OK? Should I make a bet on that?

372
00:36:38,610 --> 00:36:42,960
So I'm going to set that up. And again, this is a thing which is very hard to predict.

373
00:36:42,960 --> 00:36:46,410
So here's a little exercise we did in the lab.

374
00:36:46,410 --> 00:36:54,630
So I want you there are three magnets in this case. Can you predict as the behaviour carries on which magnets the pendulum will end up at?

375
00:36:54,630 --> 00:37:00,720
So here it goes. Often goes dancing around someone's schizophrenic and just doesn't call make up its mind where it's going to go.

376
00:37:00,720 --> 00:37:04,950
Is it going to go top or bottom? And then right at the last minute pulls down to the bottom.

377
00:37:04,950 --> 00:37:11,850
OK, so what's the prediction? There's no way it's always right as well.

378
00:37:11,850 --> 00:37:19,860
OK, so I won't make my bets on all this to winning the opening. So this is again, this is very sensitive to small changes.

379
00:37:19,860 --> 00:37:26,150
So for example, I did an experiment and I started the pendulum off top left hand corner.

380
00:37:26,150 --> 00:37:30,120
The first time I ran it. So I've coloured the magnets yellow, red and blue.

381
00:37:30,120 --> 00:37:34,830
First time it just ended up at the blue magnets. Then I changed the sixth decimal place.

382
00:37:34,830 --> 00:37:40,020
One of the coordinates of the starting point of the pendulum ran it again.

383
00:37:40,020 --> 00:37:47,190
To start with, it was fairly similar, but where it quickly went off in a completely different direction and ended up at the ridge magnets.

384
00:37:47,190 --> 00:37:52,980
I changed the six decimal place again. You can barely see any difference on the screen.

385
00:37:52,980 --> 00:37:57,300
This time it ended up with the yellow magnet so incredibly sensitive.

386
00:37:57,300 --> 00:38:06,240
And actually, this is very similar to, for example, three planets and an asteroid coming in and one of these planets is going to get wiped out.

387
00:38:06,240 --> 00:38:11,730
And you're quite like to know in advance whether it's your planet and you need to shift off or whether it's the neighbouring planet.

388
00:38:11,730 --> 00:38:21,150
But this shows you that trying to predict something is kind of important is whether are we going to get wiped out by using mathematics to predict the

389
00:38:21,150 --> 00:38:29,510
path of an asteroid is going to be very difficult because there are circumstances when that asteroid knowing the sixth decimal place of its position,

390
00:38:29,510 --> 00:38:37,740
well, pretty detailed information that might be very difficult to get. A small inaccuracy could mean that you're going to be saved or wiped out.

391
00:38:37,740 --> 00:38:42,870
And here's a graphic which helps you to to make some predictions or be able to tell you when

392
00:38:42,870 --> 00:38:47,560
you can't make predictions because it's all like mathematics is hopeless at this task.

393
00:38:47,560 --> 00:38:48,780
I mean, after all,

394
00:38:48,780 --> 00:38:56,880
we've done some pretty amazing things like landing spaceships on the science of comments as they're racing through the the Solar System,

395
00:38:56,880 --> 00:39:03,240
so we can use these powers of prediction to be able to tell quite accurately where something might go.

396
00:39:03,240 --> 00:39:10,620
But we need to know when we're in a region which has not so much sensitivity to a little bit of perturbation.

397
00:39:10,620 --> 00:39:15,300
And when we've got into our region where we're going to go wrong, we don't know what's going to happen next.

398
00:39:15,300 --> 00:39:21,390
So you see, how do you read this picture if you start the pendulum over a red region?

399
00:39:21,390 --> 00:39:26,940
It means that it's going to end up at the red magnet. So, for example, that big red blob there?

400
00:39:26,940 --> 00:39:32,940
Well, that's the way the right magnet is. So if I'm near the red magnets, the other two magnets haven't got any strength to attracted it,

401
00:39:32,940 --> 00:39:38,800
so it's just going to wobble and settle at the red magnet. So a very predictable region there.

402
00:39:38,800 --> 00:39:46,620
But look, if we go to the other side of the pendulum opposite the red magnet, again, there's a nice big swathe of rent there.

403
00:39:46,620 --> 00:39:52,500
And again, it doesn't matter a little wobble there it. It'll still the pendulum will swing backwards and forwards.

404
00:39:52,500 --> 00:39:57,870
And the other two magnets aren't strong enough to attract it, and it will just settle again, a direct magnet.

405
00:39:57,870 --> 00:40:03,720
So again, another predictable region. You can have a bit of inaccuracy and you'll still know what's going to happen next.

406
00:40:03,720 --> 00:40:11,280
But I still wanted my pendulum in the top left hand corner, and this is an area, a geometry that we call a fractal.

407
00:40:11,280 --> 00:40:17,040
Now you may have heard of fractals. Some of parents here might have gone clubbing in the 80s and danced to fractals.

408
00:40:17,040 --> 00:40:26,280
These are our shapes as you zoom in on them, the kind of psychedelic shapes that as you zoom in on them, they never seem to simplify.

409
00:40:26,280 --> 00:40:35,160
And so they kind of have this infinite kind of kind of structure to them, and their complexity remains using Zoom in.

410
00:40:35,160 --> 00:40:42,870
Something should simplify. But this is an area, for example, you might hope that, OK, it seems to be a lot of red, yellow and blue here,

411
00:40:42,870 --> 00:40:47,130
but maybe I need to get beyond the six decimal place, so I'm going to go to the seventh decimal point.

412
00:40:47,130 --> 00:40:53,660
So let's focus in on that. And maybe it'll all go yellow and I can make a prediction so I know how accurate I need to be.

413
00:40:53,660 --> 00:40:57,860
But as you zoom in, you find that no, it's still still red, yellow and blue there.

414
00:40:57,860 --> 00:41:06,230
OK, I've got to focus in a bit more. And however, for you focussing however much detail you get, the region never has a single colour.

415
00:41:06,230 --> 00:41:12,680
It always has three colours. This is the quality of a fractal that it has its infinite complexity,

416
00:41:12,680 --> 00:41:16,880
which means that in this area, it doesn't matter how accurate your measurements are.

417
00:41:16,880 --> 00:41:22,250
There's always going to be a decimal place further down, which will shift it between the three colours.

418
00:41:22,250 --> 00:41:27,290
So here you're being told that this is an area that you cannot know what's going to happen next.

419
00:41:27,290 --> 00:41:34,370
So this is very important with a dynamical system. Did you understand where can I make predictions and when do I have to say

420
00:41:34,370 --> 00:41:38,660
mathematics has the limitations now and we can't know what's going to happen next?

421
00:41:38,660 --> 00:41:42,260
This is affecting things like planetary motion, but unfortunately,

422
00:41:42,260 --> 00:41:47,930
these kind of equations control many things we would like to understand about our universe.

423
00:41:47,930 --> 00:41:51,860
For example, the weather. So you probably heard this thing, the butterfly effect.

424
00:41:51,860 --> 00:42:00,470
We have the equations for the weather. We have a huge amount of data now about the weather across the globe.

425
00:42:00,470 --> 00:42:06,590
Surely we should be able to make predictions of the weather way into the future. I think predictions have got pretty good.

426
00:42:06,590 --> 00:42:12,480
If you look at your weather app on your phone is now it's pretty accurate up to five days.

427
00:42:12,480 --> 00:42:17,270
And in fact, I've been along did a film with the BBC where we went down to the Met Office,

428
00:42:17,270 --> 00:42:24,080
where they make these predictions and what they do is they bring all the data in and then they make little perturbations of the data.

429
00:42:24,080 --> 00:42:29,990
It changes a little bit. They run about 2000 models and you can see after the first, second, third,

430
00:42:29,990 --> 00:42:34,430
fourth, fifth day, the predictions of all of these models are pretty similar.

431
00:42:34,430 --> 00:42:41,600
But there's somehow will this kind of flip over moment that after five days, suddenly predictions kind of disappear.

432
00:42:41,600 --> 00:42:45,830
And it seems like there's a sensitivity that's beyond five days.

433
00:42:45,830 --> 00:42:53,060
We're unable to make very good predictions, which is why we don't have weather forecasts for a month's time.

434
00:42:53,060 --> 00:42:57,170
So each dynamical system very often has its own sort of timescale.

435
00:42:57,170 --> 00:42:58,460
So what about the universe?

436
00:42:58,460 --> 00:43:05,120
We can make good predictions about the universe for the next five days, probably five million years, probably five billion years,

437
00:43:05,120 --> 00:43:13,070
but in five billion years, we know that these chaotic effects can start to take to start to kick in.

438
00:43:13,070 --> 00:43:18,590
And models that we've run of the universe show that there's a five percent chance that the

439
00:43:18,590 --> 00:43:24,140
the orbit of Mercury is going to start to do something really weird intersect with Venus,

440
00:43:24,140 --> 00:43:31,580
not Venus. Ouch, which will then knock the Earth out and the whole Solar System is going to fly apart like that animation that I showed you.

441
00:43:31,580 --> 00:43:35,150
But fortunately, that's not a five day scale that's on a five billion year scale.

442
00:43:35,150 --> 00:43:38,990
So the Sun will already envelop enveloped, as I think, by that point.

443
00:43:38,990 --> 00:43:44,330
So these things equations control so much that we would like to know about our universe,

444
00:43:44,330 --> 00:43:54,290
then control things like the markets seem to have an element of these chaotic equations in which means it's that very often hard to predict,

445
00:43:54,290 --> 00:43:58,910
although sometimes they're in regions that we can predict, which is why people make money in the markets.

446
00:43:58,910 --> 00:44:03,590
But the area where we see this and this is in population dynamics.

447
00:44:03,590 --> 00:44:10,310
So here's a little challenge for you. We these animals has a very strange behaviour every four years.

448
00:44:10,310 --> 00:44:17,960
This animal runs to the side of a cliff, and then they all throw themselves over in a mass suicide pact.

449
00:44:17,960 --> 00:44:20,630
So I'm going to do a little vote with you.

450
00:44:20,630 --> 00:44:27,470
So put your hands up if you think it's the muskrats which have this tendency to run to cliff faces and throw themselves off.

451
00:44:27,470 --> 00:44:31,290
OK. Quite a good little cohort of muskrats, friends here.

452
00:44:31,290 --> 00:44:36,260
OK, who thinks it's the voles that have this very strange behaviour?

453
00:44:36,260 --> 00:44:43,790
A few votes for those and and who thinks it's the lemmings that have this wow look like it's all hands go up, so a lot of votes for lemmings.

454
00:44:43,790 --> 00:44:50,570
And yes, indeed, it's the theory is that these the story is that these lemmings throw themselves over the side.

455
00:44:50,570 --> 00:44:54,980
In fact, many of you probably they put their hand up. I think I'll put my hand up as well.

456
00:44:54,980 --> 00:44:57,710
So we call that behaving like a lemming, which is, you know,

457
00:44:57,710 --> 00:45:02,850
you just follow the crowd as you one of my favourite computer games as a kid is going to show how old I am.

458
00:45:02,850 --> 00:45:08,240
That was called lemmings and you had to lead this lead lemming along and all of the others would follow.

459
00:45:08,240 --> 00:45:13,610
And you had to basically get it to fall over the side of the cliff. You see, all people know Harris shaking their heads.

460
00:45:13,610 --> 00:45:21,860
I remember that one who said, yes, indeed, lemmings, I do. It seems every four years kind of the population of lemmings disappears.

461
00:45:21,860 --> 00:45:25,490
And this theory came out that they were kind of throwing themselves over cliffs.

462
00:45:25,490 --> 00:45:31,070
And it seemed like the theory of being confirmed when Walt Disney used to make nature programmes.

463
00:45:31,070 --> 00:45:39,530
And they made a programme about the Arctic, and they managed to record these lemmings in the Arctic.

464
00:45:39,530 --> 00:45:44,200
Are they coming to the side of a cliff? And then very dramatically.

465
00:45:44,200 --> 00:45:47,810
You think he comes inside of a cliff? Most people would turn back saying, no way here.

466
00:45:47,810 --> 00:45:54,320
But now these lemmings in a massive great drop, hare suddenly started flying, flying over the side of the cliff.

467
00:45:54,320 --> 00:46:01,460
And so the crew were really pleased to get this footage, finally proving that the lemmings had this strange behaviour.

468
00:46:01,460 --> 00:46:07,130
And of course, there's no going back. But one of them is saying the guys, this is a really bad idea to turn back.

469
00:46:07,130 --> 00:46:18,380
But now over he goes as well. So it seemed like Walt Disney had proved that these lemmings behave like this a few years ago.

470
00:46:18,380 --> 00:46:22,790
The cameraman who filmed this sequence came clean.

471
00:46:22,790 --> 00:46:31,340
Those lemmings did not want to go over that cliff. In fact, the production crew had set up a spinning disc, which you couldn't see,

472
00:46:31,340 --> 00:46:37,490
and somebody in the crew was putting the lemmings on this spinning disc and they were being thrown over the side of the cliff.

473
00:46:37,490 --> 00:46:43,890
Absolutely appalling. So, so it seems like this theory hadn't got proof yet.

474
00:46:43,890 --> 00:46:50,480
Still, there lemmings every four years just die out, so we had to come up with some theory.

475
00:46:50,480 --> 00:46:56,330
It turns out it isn't, you know, call me a film crew going every four years and setting this thing up.

476
00:46:56,330 --> 00:47:02,510
So it turns out it's not lemmings suicide pact or the film crew. It's mathematics that's killing the lemmings.

477
00:47:02,510 --> 00:47:06,440
Now, many of you might say, Yeah, I'm deadly subject, but no, actually,

478
00:47:06,440 --> 00:47:12,860
there's a formula which actually controls the dynamic of the number of lemmings from one season to the next.

479
00:47:12,860 --> 00:47:16,040
So I'm going to do a little experiment to demonstrate this formula.

480
00:47:16,040 --> 00:47:22,700
So this formula in the lemmings, for example, you go in lemmings in the first season and then they they let,

481
00:47:22,700 --> 00:47:28,430
let's suppose they double up each season so they the number of lemmings increases by doubling.

482
00:47:28,430 --> 00:47:31,550
But not all of the lemmings will survive because there wasn't enough food for them.

483
00:47:31,550 --> 00:47:35,030
So the formula that we're going to use is you take the previous generation,

484
00:47:35,030 --> 00:47:39,680
multiply it by the current generation, divide by 10 and that number of lemmings won't survive.

485
00:47:39,680 --> 00:47:44,180
And this is the dynamic. Lemmings have a particular growth rate and there isn't enough food.

486
00:47:44,180 --> 00:47:48,020
So this is the kind of model that controls the number of lemmings.

487
00:47:48,020 --> 00:47:52,430
So I'm going to do a little experiment to do this experiment. I need to volunteer. So to be my lemmings.

488
00:47:52,430 --> 00:47:56,380
Yes. Good. Excellent. We need to girls go with, yeah, great.

489
00:47:56,380 --> 00:48:01,880
It doesn't work. I'm sorry. We have two boys. It just doesn't get going. So no, we're not going to do anything for real here.

490
00:48:01,880 --> 00:48:05,870
It's just theoretical. It's fine. So here are two lemmings.

491
00:48:05,870 --> 00:48:09,920
I'm good now. So the next generation, they double up and they have two more time.

492
00:48:09,920 --> 00:48:13,280
Now you can come and play. Excellent. So I'm going to need all of them.

493
00:48:13,280 --> 00:48:21,350
Yes, good. So can you come up? So now we got four lemmings, but these lemmings and they're not all going to survive because there isn't enough food.

494
00:48:21,350 --> 00:48:25,610
OK, so actually, I'm going to take you up on the stage here because it's too tempting to.

495
00:48:25,610 --> 00:48:32,600
So if you follow me up on the stage, so good. So we got to sort out which of these lemmings is not going to survive.

496
00:48:32,600 --> 00:48:36,770
So four of them in this season, but so we do the formula.

497
00:48:36,770 --> 00:48:41,720
So two times four is each divide by 10. That's roughly one is not going to survive.

498
00:48:41,720 --> 00:48:45,650
OK, so to sort this out, three cheers. That's very convenient.

499
00:48:45,650 --> 00:48:49,010
Excellent. So let's make a little bit of symmetry here.

500
00:48:49,010 --> 00:48:56,680
OK, so for lemmings, three chairs, you've got to fight for the chairs to see who survives and who goes over the cliff.

501
00:48:56,680 --> 00:49:03,890
OK. No, no, no, no, no, no, no. I like this sort of fight for survival is already great,

502
00:49:03,890 --> 00:49:12,830
but we're going to have a little bit of music to get you going because so now I have I haven't done a health and safety for this.

503
00:49:12,830 --> 00:49:18,620
So please do not get so enthusiastic that you tip over the sides here and kill you.

504
00:49:18,620 --> 00:49:23,600
So for you. OK, so we're going to have a little bit of lemming music.

505
00:49:23,600 --> 00:49:28,520
OK, and once the music stops, obviously, then you can go in for the chance and we'll say use of ice.

506
00:49:28,520 --> 00:49:34,580
So let's cue the lemming music now. I think we can pump that up a bit.

507
00:49:34,580 --> 00:49:42,790
So I will survive. The three are going to survive better. No hovering, OK, you can get rid of lemming dancing, that's it helps folks out.

508
00:49:42,790 --> 00:49:49,810
They're being very sort of on. How does the right thing?

509
00:49:49,810 --> 00:49:53,950
Oh, OK, he's all right, but I think he's got it.

510
00:49:53,950 --> 00:49:59,260
Yeah, I think being tall in this game doesn't help. So. So I'm afraid we're going to kick you over the side of the cliff.

511
00:49:59,260 --> 00:50:04,240
Oh my gosh. You can't go. I'm not going to pay for the broken limbs.

512
00:50:04,240 --> 00:50:09,340
But yeah, that's quite far, actually. OK, so we've got three surviving living, so they double up.

513
00:50:09,340 --> 00:50:14,350
So next season they have six lemmings, so I need three more volunteers, please.

514
00:50:14,350 --> 00:50:18,550
Yeah. Have you had the rank? You can come excellence and somebody on this side?

515
00:50:18,550 --> 00:50:25,900
Yes. Why don't you come across as a brave person there? That's a grown up, Josh. Easier to pick on.

516
00:50:25,900 --> 00:50:29,500
I love you. So the young the grown ups are quite keen to excellent.

517
00:50:29,500 --> 00:50:38,290
So if you come up here, so we got to if you stand up, so. So we've got six lemmings and how are you going to survive?

518
00:50:38,290 --> 00:50:43,720
Well, remember the for me to say its current generation six times three, that's 18 divide by 10.

519
00:50:43,720 --> 00:50:49,750
So that's roughly two which are not going to survive. So we need another chair in there because four four will survive.

520
00:50:49,750 --> 00:50:53,230
So let's make a little square here.

521
00:50:53,230 --> 00:50:57,850
Excellent. So for we survive, but two will not. So let's see who's going to survive this round says.

522
00:50:57,850 --> 00:51:05,260
Cuz some learning music and another one bites the dust and I too are going to bite the dust this time.

523
00:51:05,260 --> 00:51:09,880
So which two will bite the guys another one? Yeah, they've get some colouring guy.

524
00:51:09,880 --> 00:51:18,800
The I can say the same page, so we'll win with all adults being feisty and fight for survival.

525
00:51:18,800 --> 00:51:24,920
Was far too polite. After all, there's a child inside now.

526
00:51:24,920 --> 00:51:29,900
OK, so needs to get kicked over the side of the cliff and we got four survivors, though we even got grandma here.

527
00:51:29,900 --> 00:51:35,300
She said. I now think a grandmother. Yes, exactly. So in this season, so great.

528
00:51:35,300 --> 00:51:39,590
So now four doubles up to eight. So I need for more volunteers.

529
00:51:39,590 --> 00:51:43,700
So great. Yes, you can come. You can come right at the back.

530
00:51:43,700 --> 00:51:50,300
You can come. And another adult, a teacher, maybe who's brave enough to.

531
00:51:50,300 --> 00:51:54,920
Yes, exactly. Great. You can bring your laptop off and try and calculate your way out of this thing you want.

532
00:51:54,920 --> 00:52:00,290
But excellent. So we got four now. Great.

533
00:52:00,290 --> 00:52:03,920
We so we've got some some women here, so young. So four times.

534
00:52:03,920 --> 00:52:11,450
So now going eight, eight times for thirty two to five by 10, that's three and not going to survive and five will.

535
00:52:11,450 --> 00:52:15,380
So I need another chair in there. Excellent. Great as I bring that up.

536
00:52:15,380 --> 00:52:22,560
So now it's a real sort of mayhem here, a lemming party. So we need a bit more music, says cue the living music to try.

537
00:52:22,560 --> 00:52:28,800
Death triangle has decided no one else will have to guess.

538
00:52:28,800 --> 00:52:33,720
Now to the Magic three will.

539
00:52:33,720 --> 00:52:43,560
My bill. Oh, OK, so we got to who's going to go here?

540
00:52:43,560 --> 00:52:46,830
Wow, that's that's I think we have to do rock, paper scissors to decide this one.

541
00:52:46,830 --> 00:52:52,020
So OK, so you go three to one. OK, three two one show.

542
00:52:52,020 --> 00:52:55,770
OK, you can't. Yes, sir, I want to go right on this one.

543
00:52:55,770 --> 00:52:59,670
Watch out for her. She survived the whole lot. She's amazing.

544
00:52:59,670 --> 00:53:03,510
I'm great. This is great, grandma. Yeah, you don't get past me.

545
00:53:03,510 --> 00:53:12,390
OK, so but something really interesting happens now because we've got five surviving lemmings, but five double up to 10 five times.

546
00:53:12,390 --> 00:53:18,000
10 divide by 10 is five survived. Five Now die out of the 10 survive survive.

547
00:53:18,000 --> 00:53:22,830
So at this point, although the lemming generation has been growing, we now get a stability.

548
00:53:22,830 --> 00:53:30,130
So with this formula of doubling, what happens now is the lemming population stabilises at these five lemmings, so.

549
00:53:30,130 --> 00:53:35,180
So give our five lemmings a big round of applause for surviving this far and you can go back down again.

550
00:53:35,180 --> 00:53:45,720
You. What if we change the formula a little bit, so let's make the let's make the lemmings a little bit more frisky.

551
00:53:45,720 --> 00:53:50,690
So they're having a few more children. And so now they're tripling each season.

552
00:53:50,690 --> 00:53:56,490
A different behaviour kicks in because now if I did the same game again started with two lemmings,

553
00:53:56,490 --> 00:54:00,210
we get five surviving, put the five into the formula. We get eight.

554
00:54:00,210 --> 00:54:05,190
But then you get the population ping pong going backwards and forwards between five and eight.

555
00:54:05,190 --> 00:54:12,810
So instead of stabilising when the population triples, we get two values coming in sort of large and a small.

556
00:54:12,810 --> 00:54:17,940
If we gain them even more frisky, so now we're multiplying the generations by three point five.

557
00:54:17,940 --> 00:54:24,600
This is why we see four different values and this is actually lemmings are a pretty frisky kind of animal,

558
00:54:24,600 --> 00:54:32,640
and the population growth rate is about three point five, which is why every four years you see one of the values is very low.

559
00:54:32,640 --> 00:54:36,090
And this is the reason the population is being is.

560
00:54:36,090 --> 00:54:41,910
Being knocked out every four years is because of the mathematical dynamics of this formula,

561
00:54:41,910 --> 00:54:47,790
not because of some suicide pact or that film crew, but so this is still very predictable.

562
00:54:47,790 --> 00:54:53,970
But if you have a population of lemmings which is hyper frisky and then quadruple every season,

563
00:54:53,970 --> 00:54:59,010
then you see this pattern suddenly disappear and we see chaos kicking in.

564
00:54:59,010 --> 00:55:05,400
So again, we see this moment where something which was predictable suddenly becomes unpredictable, like the pendulum.

565
00:55:05,400 --> 00:55:12,210
So in this case, when we quadruple the population now, a very small change might throw one more lemming into the population.

566
00:55:12,210 --> 00:55:17,370
It can completely change the dynamics. Here you see the population completely plummet.

567
00:55:17,370 --> 00:55:25,320
You might think that something environmental or perhaps human involvement, but it's just the result of that mathematical formula.

568
00:55:25,320 --> 00:55:30,690
And this change from chaos to predictability comes into a lot of different systems.

569
00:55:30,690 --> 00:55:35,250
There's one in particular which was exploited by Roberto Carlos in an amazing way.

570
00:55:35,250 --> 00:55:39,510
And one of the greatest free kicks I think that's ever been taken in the history of football.

571
00:55:39,510 --> 00:55:43,410
This is Roberto Carlos playing for Brazil against France, but it is.

572
00:55:43,410 --> 00:55:47,640
You can't even see the goalkeeper because it's so far out and they put a wall up.

573
00:55:47,640 --> 00:55:55,290
But you think what's the point of putting a wall when you're so far out? Roberto Carlos gives it such a whack, and but Barty's doesn't move at all.

574
00:55:55,290 --> 00:55:59,040
If you see, I'm going to show you another perspective which shows you why he thought this is going way.

575
00:55:59,040 --> 00:56:02,970
Why? Because the ball stars by going way out to the right.

576
00:56:02,970 --> 00:56:06,900
But then at the last minute suddenly spins in and hits the goal in the left.

577
00:56:06,900 --> 00:56:13,770
This is the best one because you can see some of the spectators kind of ducking. And then right in the last minute, this ball swings in.

578
00:56:13,770 --> 00:56:17,100
The key to this behaviour. It's a similar thing.

579
00:56:17,100 --> 00:56:24,930
If you look at the the turbulence behind a football at high speed, the turbulence is actually chaotic.

580
00:56:24,930 --> 00:56:27,570
And this doesn't actually cause very much drag.

581
00:56:27,570 --> 00:56:33,750
So it's like if you play golf at all, I'm a golf ball can seem to always be frictionless, flying through the air.

582
00:56:33,750 --> 00:56:39,720
The little dimples in the ball mean that there's hardly any turbulence slamming the ball down.

583
00:56:39,720 --> 00:56:42,570
But there's some point at which the football passes,

584
00:56:42,570 --> 00:56:50,520
a speed at which the turbulence suddenly changes from being this chaotic thing to very predictable what we call laminar flow.

585
00:56:50,520 --> 00:56:56,190
This causes sudden drag on the football, which causes the spin to take into effect,

586
00:56:56,190 --> 00:56:59,910
and then the ball swings very dramatically into the back of the net.

587
00:56:59,910 --> 00:57:07,140
So Roberto Carlos, as well as being having been a great footballer, is clearly one of the best mathematicians we have on the planet.

588
00:57:07,140 --> 00:57:12,090
Because these equations to understand the turbulence, I mean, even me sitting at night sort of working this out.

589
00:57:12,090 --> 00:57:18,120
Yes, I think I need to do the distance to be that far away from the goal, and you need it to be quite far away.

590
00:57:18,120 --> 00:57:23,730
He was able to to make this thing work. In fact, this is another of our millennium prises.

591
00:57:23,730 --> 00:57:30,180
This is another of the no mysteries. We do not understand the equations behind the turbulence of a football and well

592
00:57:30,180 --> 00:57:34,980
enough to be able to to write down the equations of really what this is doing.

593
00:57:34,980 --> 00:57:39,630
So I hope that showing you that mathematics is a very powerful tool to help to make predictions.

594
00:57:39,630 --> 00:57:46,830
But there are limitations to this. And if you want to point out a little bit more about these stories, some of the books that Alan mentioned,

595
00:57:46,830 --> 00:57:51,100
the number of mysteries is actually a book that I wrote after doing the Christmas lectures.

596
00:57:51,100 --> 00:57:57,150
The royal institution. Talking about these I've got with me a few books of what we cannot know,

597
00:57:57,150 --> 00:58:01,410
which shows you the limitations not just in maths, but in science more generally.

598
00:58:01,410 --> 00:58:07,110
In fact, they're very special editions that I brought you because they're actually American editions, which you can't get hold of here.

599
00:58:07,110 --> 00:58:11,460
The Americans call the book the Great Unknown rather than what we cannot know,

600
00:58:11,460 --> 00:58:18,030
but those are on sale afterwards and and do look out also for the forthcoming book, which is the creativity code.

601
00:58:18,030 --> 00:58:24,630
So this is the proof copy that I've got, which is talking about the the power of ITER to perhaps predict the future.

602
00:58:24,630 --> 00:58:48,249
But thank you for all your contributions and attention.