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.
I think we, as humans spend a lot of our time thinking about the future.
I did an exercise through the BBC. We were doing a programme about consciousness,
and I think it's one of the unique kind of traits of consciousness that it allows us to imagine ourselves into the future.
I have a cat at home and my cat just sits there enjoying the day.
I don't think it spends very much time thinking about what's going to happen next,
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.
I think his on subject of mathematics actually is very curious.
When I'm at a party and somebody asks me, What do you do? And I say, I'm a mathematician.
I don't know whether you find this alone,
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,
very quickly and I before they go. They always seem to tell me what they got in the sea level.
I'm not sure what that's about, but anyway,
I'm very persistent and I chased after them and I began to realise that mathematicians were very misunderstood breed.
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,
I've got this feeling that they think I'm doing long division to a lot of decimal places.
And, you know, surely my computer has put me out of a job by now.
But I tried to explain to them that no, actually mathematics.
If I was going to define mathematics, I would define it as the science of pattern searching.
And it's not ability to understand a pattern which gives us this great tool to be able to look into the future.
Because by looking at a pattern of data in the past, if we can find that pattern,
then we might be able to read that pattern so we can predict what it's going to do in the future.
How do we know about so what's the trajectory of our universities?
Because we've seen the data of the universe from the telescopes that we've seen, it's amazing.
We're stuck on this planet. Yet through the information we've got,
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,
even on our smaller, more local scale of the universe by looking at the data of the climate.
We're beginning to understand what might be happening in the future.
We're going to have to use our science and mathematics to actually do something about what that trajectory looks like.
So I think for me, mathematics is all about looking for patterns.
So what I would love to do in this presentation is to show you some of the power of mathematics to predict things,
but also some of the limitations that mathematics can't let us know everything about the future.
I think that time the challenge of patent searching for me is kind of perfectly encapsulated in the
challenge you probably had at school where you're given a sequence of numbers and you have to try and spot,
well, what's the logic behind this sequence of numbers to be able to tell what's going to happen next?
So I thought I'd see how good you are here in Solihull at predicting what's going to happen next.
So I've got a series of challenges for you, which gets a little bit more difficult as we go on.
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?
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?
Thirty six. Very easy. So how did you get thirty six?
Yeah, you know what I'm pointing you use see, we're very loud and between the first two is two and three four.
Exactly. You're sort of adding one more number on each time one plus two plus three plus four.
In fact, we call these the triangular numbers because you can view them very geometrically its number of stones.
You need to make a triangle. How many stones you need to when you add on an extra layer each time.
And actually, if you wanted to know, for example, what the hundredth triangular number is,
so how would you know way into the future what this number sequence is going to do?
Well, you could do the kind of the dumb way would to be add up all the numbers from one to 100.
But mathematicians, we're not very good at kind of arithmetic, actually. So I'd probably make a mistake if I did that.
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.
And here's a trick I often use in my own research, which is kind of changing the language.
So instead of considering numbers to get this formula, if you consider these things more geometrically, here's some triangles.
If I put two triangles together, I get a rectangle.
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.
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.
So this is a sequence we understand very well.
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.
First challenges Dan Brown sets you in the da Vinci Code. So what's the number in this sequence?
One one two three five eight 13 21. Thirty four.
OK, a very famous sequence. You got that by adding the two previous numbers together.
So these are called the Fibonacci numbers named after this Italian mathematician in the 12th 13th century,
who understood that these are nature's favourite numbers. You find them all over the natural world.
So for example, if you count a fruit, open an apple sort of horizontally, you find a five pointed star inside or a banana.
You'll find a three pointed star, a Sharon fruit,
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.
Invariably, it's a number in the Fibonacci sequence. And if it isn't the number in the Fibonacci sequence,
that's because a petal has fallen off your flower, which is how mathematicians get round exceptions.
And so actually, these numbers shouldn't be called the Fibonacci numbers.
I discovered quite recently that they were discovered before Fibonacci ever found them in nature, not by mathematicians,
but by Indian musicians and poets who understood that these numbers also help you to count the number
of rhythms that are possible if you're a tabla player that you can make out of long and short beats.
So the wonderful thing is you find mathematics all over the place, not just nature, but also in the world of art as well.
Now this sequence, we learnt quite a lot about this sequence.
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.
It's a bit more complicated than the triangular numbers involve this one phone number, the the golden ratio.
You have to take powers of the golden ratio, but there's a lot of mysteries we still don't know about these numbers.
So there's still some things no mysteries to solve about the Fibonacci numbers.
OK, the next sequence some one two four eight 16.
Thirty two is a possible answer, but thirty one is also an equally valid answer to the sequence.
You mean it's OK, this is just doubling.
Thirty two seems to be the most obvious, but why if any one shot out thirty one, would that be equally correct?
Well, these numbers up to 16, are actually counting something called the circle division numbers.
So what are these? If only take a circle. And I took one point in that circle.
I just got one region, the circle in the middle. But now let's pick another point and join these two points up.
Now I've got two regions, so let's keep on going well and another points.
Join up all the lines. I've got a triangle in the middle, three regions around the outside, that's four regions.
OK, so you think, OK, well, I'm starting to spot the pattern here at a point.
Another one. We join all of the four points up. Well, this nice little envelope in the middle for triangles there.
Four regions round the outside. Eight you keep going.
What about five? Sure enough, you get this lovely five pointed star Pentagon 16.
And by now you think, Oh, it's pretty obvious. Every time you add a points, you somehow double up the number of regions.
And so you make a conjecture that the pattern is going to go in this direction, just double.
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.
So for me, this is quite one of the assignments of mathematics because they're often kind of surprises out there.
Unexpected directions that something can go in. So there is a formula for the number of regions.
It's not tumbling. It's a slightly more complicated formula.
It's this cortical polynomial way you raise the number of points on the circle to the power four and do a further calculation.
But this formula tells us how this number of regions grows and grows much slower than just doubling.
And this might be a practical problem where you're actually got some sort of
network and you need to know how many regions this can be divided up into.
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.
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.
So you have to be prepared for surprises in mathematics, which is one of the reasons it's so much fun in a way.
OK, so oh, and then a little bit more. That's that spoils my OK.
So if anybody saw quite what popped out there, a little bit more challenging sequence two nine, 10 11 13 16.
Usually, this has the mathematicians really working hard because they see some sequence going here,
but as as I fleshed out, probably too quickly there.
These don't have any pattern at all because these are in fact, the National Lottery winning numbers for the 28th of September.
So another warning here that not everything has a pattern to it.
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,
but which actually just the result of kind of randomness in the pattern can kind of just disappear.
But in a little bit, I'll show you that even when something is as random as the National Lottery,
mathematics has tools to be able to make quite powerful predictions about even things which are genuinely random.
OK, what about the last sequence? Two three five seven 11 13 17 19?
Twenty three, exactly, and these are the the prime numbers.
So these are probably the most important numbers in the whole of mathematics,
the invisible numbers, the numbers which can't be divided by anything except themselves.
And one and I would say of all of the numbers in mathematics,
these are the most important because these are literally the building blocks of the whole of our subject.
One take a number like one hundred and forty five. Is that a prime number now clearly divisible by five?
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.
But now you see our divided and divided, and now I've got down to three numbers which are indivisible.
These can't be divided any further. And these are the building blocks which make the number one hundred and five.
So this is true of any number. You take your telephone number, for example, it might already be a prime number,
in which case I'm incredibly envious because I've always wanted a prime number telephone number.
I've got an even telephone number. It's completely rubbish. But so I will swap with you if you have a prime number.
But if not, here's a challenge for you.
Go back and actually keep on dividing it until you get down to the primes which built which build your telephone number.
And so every number is built out of these primes.
For me from 30th magicians, these numbers are really like the hydrogen and oxygen of the world of mathematics.
Probably the most important thing in chemistry. Anybody who's had chemistry is today at school.
They will have had the periodic table up on the wall because that is the ingredients from which our universe is made.
And Mendel have spotted a pattern in these atoms, which enabled him to make predictions about atoms that were missing from this list.
So that's the power of spotting patterns.
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.
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.
These numbers, we know, go on forever. There are infinitely many of them.
However, this is the largest prime number we've found so far.
I mean, it's pretty big, too, to the panel of seventy seven million two hundred thirty two thousand nine hundred seventeen.
Well, that's very divisible. We take one off. That's no. You get a prime number.
This has over twenty three million digits. Bernie Bink number.
If I said all the digits out loud, I think I calculated that we'd be here for about three months.
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
asked you for a billion digit triangular number or a billion digit Fibonacci number.
Then we could use the formulas that we found to be able to generate those quite quickly.
But prime numbers, we don't understand a formula to generate primes.
We don't understand a pattern which will help us to to find them quickly.
So although there are infinitely many of these numbers, they still represent one of the greatest mysteries.
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.
So it's one of the things that I work on in my research in Oxford is trying
to understand these numbers because they are so fundamental to our subject.
So hopefully,
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.
If you want to find perhaps the next biggest prime number, this was actually not discovered by a professional mathematician,
but by an amateur who downloaded a piece of software onto his computer.
And the software uses a little algorithm that we've discovered, which helps to find these special primes called Amazon Prime's.
And there's actually prises, I think, $3000 for the next biggest prise that you discover if you pass the 100 million digit mark.
There's a hundred and fifty thousand dollar prise for that and a billion digits, so you can get one is bigger than that.
We know it exists because of the proof for the ancient Greeks. There's a two hundred thousand dollar prise for that one.
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.
But if you can understand the true mystery of these numbers, how they are laid out in the universe of numbers,
in fact, that's one of the things that we call one of our millennium prises.
There is a million dollar prise offered by a businessman in America.
For anybody, you can truly understand the mystery of how the primes are laid out.
So, so these are really one of our biggest no mysteries in the whole of mathematics.
In fact, we think there may not be any pattern at all, which is sort of harder to prove than there is a pattern.
And we think that the primes the way they're laid out has much.
More in common with these lottery ticket numbers in the triangle or Fibonacci numbers,
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.
I mean, how do you show there's no pattern that's much more difficult than actually finding a pattern to help predict them?
But even when something doesn't have a pattern, when something's genuinely random,
still, you're able to tell some interesting things about these numbers.
So, for example, let's take the National Lottery. In fact,
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.
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.
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.
So if you've got a pen or borrow a pen from a neighbour, perhaps circle your numbers, your six choices.
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.
So a little sorry. There's no prises.
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.
Please choose your numbers now and not when they come out of the box.
I know it's very tempting to say just have a little look over with your neighbour has actually done it.
And that's another thing. Actually,
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,
very rare, as you will see. And probably if one of you does.
I would predict probably you, you were waiting for the ball to come out.
OK, so you've already chosen your numbers. You can play this actually if you're watching my Facebook Live.
Just write down some numbers six numbers between one and 49.
And let's see. Maybe actually if we've got thousands are watching online.
Somebody might actually get six, but they can tweet us live and see what they get.
OK, so are you ready? Let's get some numbers to be chosen. So if you could choose our first number, please?
Great number seven grades, a prime number of rights, yes, think of that number seven.
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.
Number three, you are, yeah, good on primes already. Yeah. Number 10, OK.
Online, we get 10. Number of digits we use on our hands.
Thirty one. Yes. Quite prime laden number one so far.
Are you going on a great, excellent number for a lot of low lying numbers here?
And finally, number thirty three. OK, so I didn't hear any Woohp say probably, no, but you got all six, right?
So let me put these in order. Oops. Come back. That's one, right?
Just put these. So we've got about three four.
It's three four. All right.
Seven, 10. OK, so I'll just read them out again.
So we've got three. Number three, number four, number seven, number 10, number 31 and number thirty three.
Great. Now about these points in a lecture.
I always get a little bit sleepy and I'm starting to drift off.
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.
That's great. Yes, I want to wave your things. Let's see all these lottery papers.
Great. I'm going to take a picture of you all say, Yeah, that's fantastic. Excellent.
I shall tweet that later. Good. Right. Keep standing. Now I'm going to make a few predictions about the numbers,
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.
So I want you to sit down. If you didn't get any numbers right at all.
And let's see whether how good the maths is. I'm making a prediction.
So I think that's I think you're a little over the half.
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.
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?
Let's see. We should. So this is an eighth of you.
So let's see. We've got about it's not about an eight standing.
I would say. That's pretty good. Yeah. So but what about three numbers?
OK, so one in 50 chance. So I would say we've got about 200. So maybe five of you will standing.
Let's make a prediction. Five of you would got three numbers or more, right?
So keep saying you've got three or more numbers, right? This would just just her.
Oh my God, you're so unlucky and so on the whole. Oh, well, that's OK.
So of course, these predictions are only in the corner. You're sitting down already.
You didn't get four, right? You only got three, right?
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.
And she was actually because four we need one in a thousand will be getting four right one in fifty five thousand eleven.
What he's maybe going fifty five thousand people watching on Facebook.
Certainly some of our lectures that we live from Oxford are getting into the sixty thousand range.
We might have one of the people watching who's got five numbers right, but six numbers.
We haven't quite hit this many audience watching online as 14 million,
but it's a one in a 14 million chance that you would get all of those six numbers right?
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.
So one in 40 million is almost like, well, that's never going to happen.
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.
So space, you bought a lottery ticket every week. How long would it take you to get four five six numbers, right?
Well, OK, you bought a lottery ticket every week after one year,
you might have three numbers one in fifty seven, but it took 20 years to get four numbers right.
So I worked out to get five numbers right. If King Alfred's have been buying lottery tickets.
He might have got five numbers right. But to get all six numbers right, I think the first Homo sapiens that ever involved,
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.
Then by now, they might have once got all six numbers, right?
So this gives you a sense of the scale of these large numbers.
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.
But the mathematics can help you to kind of work out what the chances of getting these numbers.
But what mathematics can help you to do is to avoid something rather strange that happened in the ninth
week that the National Lottery was run in the UK because in the ninth week of the National Lottery.
An amazing hundred and thirty three people won the jackpot that week.
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.
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
share the million with another one hundred and thirty two people who've won that week.
This is the worst week to win the National Lottery. So, so what happens that week?
So I mean, there's so many people we usually use two or three, one, maybe none, but one hundred and thirty three people.
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?
What happened that week is evidence of the fact that as humans, we're very bad at behaving randomly.
We tend to leave patterns wherever we go. In fact, we see a pattern rich.
This is one of the reasons that companies that have access to our data.
People like Facebook and Google are able to use the data so powerfully to make predictions about what we're likely to do next.
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.
You see, what happened that week is that very often people, when they choose numbers randomly,
they like to space the numbers out rather evenly across the page.
And this is very common. And if you see the numbers that came out of the lottery that week, they were nicely, evenly spaced.
So it means that it favoured all of those people, like two evenly spaced their numbers, which is the majority of humans.
And so when it is nicely spaced like this, it means that a lot of people are likely to win the jackpot.
So let's I'm going to ask you a question now about how randomly you behaved.
So I want you to stand up if you choose two consecutive numbers.
So seventeen eighteen, twenty one twenty two. So let's see, because those obviously aren't very evenly spaced out.
OK, so what numbers did you choose it? Six and seven.
OK, there you go. So now quite a lot of you are standing, but if you were genuinely behaving randomly, half of you,
half of you should be standing now because half of the numbers that come out of this box will have two consecutive numbers.
In fact, tonight we had three and four come out of there the previous week to the National Lottery,
the eighth week we had twenty one in 20 to 30 and 31 came out the next week.
So if you were behaving randomly, half of you should be standing.
Now I want you to sit down unless you chose the six numbers one two three four five six.
Did anyone choose one two three four five six? Let's see if anybody is standing.
What would you what? What does this mean? Why the date? What you dayton's 40, 44 45 46.
Oh OK, I can see what you're saying there. Yeah, right? Yeah, that's in the spirit of what I'm asking.
So that's fair enough. So would you choose one two three four five six? So let me.
These are the genuine mathematicians in the room,
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.
So you're probably saying, well, why don't you just one two three little five six? That's never going to come out?
Actually, we did not quite go quite close there. OK, good.
But you do choose one two three. So you can sit down. Thank you. Five.
If you don't choose that because apparently ten thousand people a week to that thing just goes to
show how intelligent the UK population is because they realise that's as likely as anything else.
So actually, what you need to do is what this guy here did,
because that's very unusual to choose the six at the end rather than the six at the beginning.
So. So this is a way to help you in the National Lottery, which is OK.
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.
So clump your numbers together. And what you want to do in any investment basically is to go against the tide.
Such a win, you win, you win big. But mathematics can also help you to work out when something looks really
bizarre and strange is actually something you would expect out of randomness.
So we've seen that clumping together is something you expect now.
Something really weird happened in the Bulgarian lottery on in 2009,
and these are the winning lottery ticket numbers on the 6th of September four fifteen twenty three twenty four two consecutive numbers came out.
Thirty five and forty two satellites, the universe and everything.
Nothing unusual there. Except that the following four days later, on the 10th of September, the same six numbers came out.
Now that looks really suspicious, especially when it's Bulgaria.
I mean, No. Three, Sorry. Anyone watching from Bulgaria, I'm not causing any aspersions on my.
So I mean, they're bringing in the. Gary in government said, Oh, come on.
Something is really going on here. There's something, you know, clearly some corruption going on.
They've run the the video twice or whatever. So they started in an investigation and I couldn't find anything weird going on.
Mathematicians got involved in this and they started to do some calculations.
We've had a lot of lotteries running across the world every week in different territories, different countries, different counties.
By now, we realise that this clumping means that this should happen at some stage,
at some stage where it should be a lottery, where six numbers come out consecutively.
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.
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.
My wife, you, for example, she thinks she's psychic because, you know, she'll think about somebody and they'll phone.
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.
But I mean, she might well be psychic donate. So that's a lot about the mathematics of randomness.
But now I want to come to an area where mathematics is really having problems about making predictions.
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.
And this is the mathematics of chaos. Chaos is a mathematical idea of chaos was first discovered by this French mathematician, Andre Grey,
who is actually trying to solve a problem that was set by the King of Sweden as a challenge for his birthday.
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.
And the challenge that you sent mathematicians for that birthday was a question about our Solar System.
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,
our Solar System might fly apart there. All the planets were just kind of do something really strange.
Now it looks at the Solar System is pretty predictable because we're really good at
making predictions about when there's going to be a transit of Venus or or an eclipse,
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.
So it looks like our Solar System is stable, but can we mathematically prove that's true?
So one can say, OK, I'm going to take our mathematical tools. It's just a bit of Newtonian physics gravity.
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,
that they will just carry on doing ellipses around each other till the end of time.
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.
But when you throw in another planet into this little solar system, very strange things start to happen.
So you can set up a very stable system where these three planets are now doing ellipses around us.
But these with three planets,
a very small change in the initial conditions of one of those planets can cause something that looks stable to suddenly fly apart.
So here these three planets seem to be starting in a similar way,
but actually on May one of this planet slightly smaller than the other ones and suddenly, after a while, these planets just fly apart.
I mean, you really don't want to be living in this solar system.
So but it was just a very small change in in the system caused the thing to behave in a completely different way.
And this is the signature of chaos because I think many, many people believed that, you know, Newton,
to come up with the laws of nature that the laws of motion and we got the equations for how the things evolve,
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.
And OK, we may not have a precise description of the universe now,
but a good approximation should at least give us a good approximation of the future.
But what's point understood is that even a very small error, as we've seen here in our description of the present,
can cause two different models to go in completely different directions and predict completely different answers.
I think my favourite example of chaos is actually a very simple system, so it doesn't, you know, this is quite simple.
It's just planets are basically just like little points. The mathematics is not complicated and it doesn't appear complicated.
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.
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.
And so the geometry of this is totally straightforward is to rectangle similar lengths.
The physics is fairly straightforward. It's just gravity.
This is made to be practically frictionless, so it doesn't really contribute anything to this.
But trying to predict what this thing is going to do next is almost impossible.
Wow. Gone, gone.
Oh, yes.
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.
I wasn't expecting it to do that.
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.
So I've got a little notch here, which I try and predict, try and repeat exactly the same behaviour.
So you remember what it just did? OK, well, maybe not spare time.
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.
Well, you see, there we go. It sort of. So it's got a little tired now.
I've got time quickly. So we've got a completely different behaviour after a while.
So that's the signature of chaos. This small change can cause a completely different behaviour.
This is my favourite desktop toy. I can play with this for absolutely hours.
So if you'll indulge me, let's do one more of those things they offer you.
Oh, oh, it's really showing off now, gosh, well, look, know that.
We are dancing back and forth of hips. Yeah, good go round.
You got any more. Now he's shaking his head, OK, I guess.
So that's actually one of my other favourite toys.
Is this one? I've got this on my desk at home. So this is again a pendulum.
And there are six magnets which attract the pendulum and overreach magnets.
It's an answer to a question. So if, for example, you can ask a question, it will answer yes.
Maybe. Definitely no way. Try again. Ask a friend. So I use this to make all my decisions in life.
So. So I'm a big Arsenal fan. So is there a chance we're going to win the Europa League this year?
That's probably all we're possibly going to do. OK? Should I make a bet on that?
So I'm going to set that up. And again, this is a thing which is very hard to predict.
So here's a little exercise we did in the lab.
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?
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.
Is it going to go top or bottom? And then right at the last minute pulls down to the bottom.
OK, so what's the prediction? There's no way it's always right as well.
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.
So for example, I did an experiment and I started the pendulum off top left hand corner.
The first time I ran it. So I've coloured the magnets yellow, red and blue.
First time it just ended up at the blue magnets. Then I changed the sixth decimal place.
One of the coordinates of the starting point of the pendulum ran it again.
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.
I changed the six decimal place again. You can barely see any difference on the screen.
This time it ended up with the yellow magnet so incredibly sensitive.
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.
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.
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
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,
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.
And here's a graphic which helps you to to make some predictions or be able to tell you when
you can't make predictions because it's all like mathematics is hopeless at this task.
I mean, after all,
we've done some pretty amazing things like landing spaceships on the science of comments as they're racing through the the Solar System,
so we can use these powers of prediction to be able to tell quite accurately where something might go.
But we need to know when we're in a region which has not so much sensitivity to a little bit of perturbation.
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.
So you see, how do you read this picture if you start the pendulum over a red region?
It means that it's going to end up at the red magnet. So, for example, that big red blob there?
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,
so it's just going to wobble and settle at the red magnet. So a very predictable region there.
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.
And again, it doesn't matter a little wobble there it. It'll still the pendulum will swing backwards and forwards.
And the other two magnets aren't strong enough to attract it, and it will just settle again, a direct magnet.
So again, another predictable region. You can have a bit of inaccuracy and you'll still know what's going to happen next.
But I still wanted my pendulum in the top left hand corner, and this is an area, a geometry that we call a fractal.
Now you may have heard of fractals. Some of parents here might have gone clubbing in the 80s and danced to fractals.
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.
And so they kind of have this infinite kind of kind of structure to them, and their complexity remains using Zoom in.
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,
but maybe I need to get beyond the six decimal place, so I'm going to go to the seventh decimal point.
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.
But as you zoom in, you find that no, it's still still red, yellow and blue there.
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.
It always has three colours. This is the quality of a fractal that it has its infinite complexity,
which means that in this area, it doesn't matter how accurate your measurements are.
There's always going to be a decimal place further down, which will shift it between the three colours.
So here you're being told that this is an area that you cannot know what's going to happen next.
So this is very important with a dynamical system. Did you understand where can I make predictions and when do I have to say
mathematics has the limitations now and we can't know what's going to happen next?
This is affecting things like planetary motion, but unfortunately,
these kind of equations control many things we would like to understand about our universe.
For example, the weather. So you probably heard this thing, the butterfly effect.
We have the equations for the weather. We have a huge amount of data now about the weather across the globe.
Surely we should be able to make predictions of the weather way into the future. I think predictions have got pretty good.
If you look at your weather app on your phone is now it's pretty accurate up to five days.
And in fact, I've been along did a film with the BBC where we went down to the Met Office,
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.
It changes a little bit. They run about 2000 models and you can see after the first, second, third,
fourth, fifth day, the predictions of all of these models are pretty similar.
But there's somehow will this kind of flip over moment that after five days, suddenly predictions kind of disappear.
And it seems like there's a sensitivity that's beyond five days.
We're unable to make very good predictions, which is why we don't have weather forecasts for a month's time.
So each dynamical system very often has its own sort of timescale.
So what about the universe?
We can make good predictions about the universe for the next five days, probably five million years, probably five billion years,
but in five billion years, we know that these chaotic effects can start to take to start to kick in.
And models that we've run of the universe show that there's a five percent chance that the
the orbit of Mercury is going to start to do something really weird intersect with Venus,
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.
But fortunately, that's not a five day scale that's on a five billion year scale.
So the Sun will already envelop enveloped, as I think, by that point.
So these things equations control so much that we would like to know about our universe,
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,
although sometimes they're in regions that we can predict, which is why people make money in the markets.
But the area where we see this and this is in population dynamics.
So here's a little challenge for you. We these animals has a very strange behaviour every four years.
This animal runs to the side of a cliff, and then they all throw themselves over in a mass suicide pact.
So I'm going to do a little vote with you.
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.
OK. Quite a good little cohort of muskrats, friends here.
OK, who thinks it's the voles that have this very strange behaviour?
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.
And yes, indeed, it's the theory is that these the story is that these lemmings throw themselves over the side.
In fact, many of you probably they put their hand up. I think I'll put my hand up as well.
So we call that behaving like a lemming, which is, you know,
you just follow the crowd as you one of my favourite computer games as a kid is going to show how old I am.
That was called lemmings and you had to lead this lead lemming along and all of the others would follow.
And you had to basically get it to fall over the side of the cliff. You see, all people know Harris shaking their heads.
I remember that one who said, yes, indeed, lemmings, I do. It seems every four years kind of the population of lemmings disappears.
And this theory came out that they were kind of throwing themselves over cliffs.
And it seemed like the theory of being confirmed when Walt Disney used to make nature programmes.
And they made a programme about the Arctic, and they managed to record these lemmings in the Arctic.
Are they coming to the side of a cliff? And then very dramatically.
You think he comes inside of a cliff? Most people would turn back saying, no way here.
But now these lemmings in a massive great drop, hare suddenly started flying, flying over the side of the cliff.
And so the crew were really pleased to get this footage, finally proving that the lemmings had this strange behaviour.
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.
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.
The cameraman who filmed this sequence came clean.
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,
and somebody in the crew was putting the lemmings on this spinning disc and they were being thrown over the side of the cliff.
Absolutely appalling. So, so it seems like this theory hadn't got proof yet.
Still, there lemmings every four years just die out, so we had to come up with some theory.
It turns out it isn't, you know, call me a film crew going every four years and setting this thing up.
So it turns out it's not lemmings suicide pact or the film crew. It's mathematics that's killing the lemmings.
Now, many of you might say, Yeah, I'm deadly subject, but no, actually,
there's a formula which actually controls the dynamic of the number of lemmings from one season to the next.
So I'm going to do a little experiment to demonstrate this formula.
So this formula in the lemmings, for example, you go in lemmings in the first season and then they they let,
let's suppose they double up each season so they the number of lemmings increases by doubling.
But not all of the lemmings will survive because there wasn't enough food for them.
So the formula that we're going to use is you take the previous generation,
multiply it by the current generation, divide by 10 and that number of lemmings won't survive.
And this is the dynamic. Lemmings have a particular growth rate and there isn't enough food.
So this is the kind of model that controls the number of lemmings.
So I'm going to do a little experiment to do this experiment. I need to volunteer. So to be my lemmings.
Yes. Good. Excellent. We need to girls go with, yeah, great.
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.
It's just theoretical. It's fine. So here are two lemmings.
I'm good now. So the next generation, they double up and they have two more time.
Now you can come and play. Excellent. So I'm going to need all of them.
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.
OK, so actually, I'm going to take you up on the stage here because it's too tempting to.
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.
So four of them in this season, but so we do the formula.
So two times four is each divide by 10. That's roughly one is not going to survive.
OK, so to sort this out, three cheers. That's very convenient.
Excellent. So let's make a little bit of symmetry here.
OK, so for lemmings, three chairs, you've got to fight for the chairs to see who survives and who goes over the cliff.
OK. No, no, no, no, no, no, no. I like this sort of fight for survival is already great,
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.
So please do not get so enthusiastic that you tip over the sides here and kill you.
So for you. OK, so we're going to have a little bit of lemming music.
OK, and once the music stops, obviously, then you can go in for the chance and we'll say use of ice.
So let's cue the lemming music now. I think we can pump that up a bit.
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.
They're being very sort of on. How does the right thing?
Oh, OK, he's all right, but I think he's got it.
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.
Oh my gosh. You can't go. I'm not going to pay for the broken limbs.
But yeah, that's quite far, actually. OK, so we've got three surviving living, so they double up.
So next season they have six lemmings, so I need three more volunteers, please.
Yeah. Have you had the rank? You can come excellence and somebody on this side?
Yes. Why don't you come across as a brave person there? That's a grown up, Josh. Easier to pick on.
I love you. So the young the grown ups are quite keen to excellent.
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?
Well, remember the for me to say its current generation six times three, that's 18 divide by 10.
So that's roughly two which are not going to survive. So we need another chair in there because four four will survive.
So let's make a little square here.
Excellent. So for we survive, but two will not. So let's see who's going to survive this round says.
Cuz some learning music and another one bites the dust and I too are going to bite the dust this time.
So which two will bite the guys another one? Yeah, they've get some colouring guy.
The I can say the same page, so we'll win with all adults being feisty and fight for survival.
Was far too polite. After all, there's a child inside now.
OK, so needs to get kicked over the side of the cliff and we got four survivors, though we even got grandma here.
She said. I now think a grandmother. Yes, exactly. So in this season, so great.
So now four doubles up to eight. So I need for more volunteers.
So great. Yes, you can come. You can come right at the back.
You can come. And another adult, a teacher, maybe who's brave enough to.
Yes, exactly. Great. You can bring your laptop off and try and calculate your way out of this thing you want.
But excellent. So we got four now. Great.
We so we've got some some women here, so young. So four times.
So now going eight, eight times for thirty two to five by 10, that's three and not going to survive and five will.
So I need another chair in there. Excellent. Great as I bring that up.
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.
Death triangle has decided no one else will have to guess.
Now to the Magic three will.
My bill. Oh, OK, so we got to who's going to go here?
Wow, that's that's I think we have to do rock, paper scissors to decide this one.
So OK, so you go three to one. OK, three two one show.
OK, you can't. Yes, sir, I want to go right on this one.
Watch out for her. She survived the whole lot. She's amazing.
I'm great. This is great, grandma. Yeah, you don't get past me.
OK, so but something really interesting happens now because we've got five surviving lemmings, but five double up to 10 five times.
10 divide by 10 is five survived. Five Now die out of the 10 survive survive.
So at this point, although the lemming generation has been growing, we now get a stability.
So with this formula of doubling, what happens now is the lemming population stabilises at these five lemmings, so.
So give our five lemmings a big round of applause for surviving this far and you can go back down again.
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.
So they're having a few more children. And so now they're tripling each season.
A different behaviour kicks in because now if I did the same game again started with two lemmings,
we get five surviving, put the five into the formula. We get eight.
But then you get the population ping pong going backwards and forwards between five and eight.
So instead of stabilising when the population triples, we get two values coming in sort of large and a small.
If we gain them even more frisky, so now we're multiplying the generations by three point five.
This is why we see four different values and this is actually lemmings are a pretty frisky kind of animal,
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.
And this is the reason the population is being is.
Being knocked out every four years is because of the mathematical dynamics of this formula,
not because of some suicide pact or that film crew, but so this is still very predictable.
But if you have a population of lemmings which is hyper frisky and then quadruple every season,
then you see this pattern suddenly disappear and we see chaos kicking in.
So again, we see this moment where something which was predictable suddenly becomes unpredictable, like the pendulum.
So in this case, when we quadruple the population now, a very small change might throw one more lemming into the population.
It can completely change the dynamics. Here you see the population completely plummet.
You might think that something environmental or perhaps human involvement, but it's just the result of that mathematical formula.
And this change from chaos to predictability comes into a lot of different systems.
There's one in particular which was exploited by Roberto Carlos in an amazing way.
And one of the greatest free kicks I think that's ever been taken in the history of football.
This is Roberto Carlos playing for Brazil against France, but it is.
You can't even see the goalkeeper because it's so far out and they put a wall up.
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.
If you see, I'm going to show you another perspective which shows you why he thought this is going way.
Why? Because the ball stars by going way out to the right.
But then at the last minute suddenly spins in and hits the goal in the left.
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.
The key to this behaviour. It's a similar thing.
If you look at the the turbulence behind a football at high speed, the turbulence is actually chaotic.
And this doesn't actually cause very much drag.
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.
The little dimples in the ball mean that there's hardly any turbulence slamming the ball down.
But there's some point at which the football passes,
a speed at which the turbulence suddenly changes from being this chaotic thing to very predictable what we call laminar flow.
This causes sudden drag on the football, which causes the spin to take into effect,
and then the ball swings very dramatically into the back of the net.
So Roberto Carlos, as well as being having been a great footballer, is clearly one of the best mathematicians we have on the planet.
Because these equations to understand the turbulence, I mean, even me sitting at night sort of working this out.
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.
He was able to to make this thing work. In fact, this is another of our millennium prises.
This is another of the no mysteries. We do not understand the equations behind the turbulence of a football and well
enough to be able to to write down the equations of really what this is doing.
So I hope that showing you that mathematics is a very powerful tool to help to make predictions.
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,
the number of mysteries is actually a book that I wrote after doing the Christmas lectures.
The royal institution. Talking about these I've got with me a few books of what we cannot know,
which shows you the limitations not just in maths, but in science more generally.
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.
The Americans call the book the Great Unknown rather than what we cannot know,
but those are on sale afterwards and and do look out also for the forthcoming book, which is the creativity code.
So this is the proof copy that I've got, which is talking about the the power of ITER to perhaps predict the future.
But thank you for all your contributions and attention.