# Horizon (1964) s47e10 Episode Script

### Alan and Marcus Go Forth and Multiply

So does the same thing apply to
light? Because if light
left the
Earth and kept going straight up
Two weeks ago I never would
have believed it was possible.

Me, Alan Davies, discussing the mathematics of light travelling through the universe on a hilltop in the middle of winter with an Oxford University professor? You have completely transformed how I see the universe, Marcus.

Yeah?! But I've just had a peculiar adventure.

It took me to the fourth dimension.

That looks like a fun bit of maths.

To the outer reaches of space.

Se we're living in a four-dimensional doughnut.

To the deepest recesses of my mind.

Look at it.

It's like the Pacific.

Bermuda Triangle maybe.

Yeah.

And to Weymouth.

Brilliant.

That is an act of genius, Alan.

To see if I can be convinced that maths is not the boring subject I hated at school, I've agreed to undertake a mathematical assault course, with this man as my guide.

Yes! It's 13, get in there! Prime number I told you.

His name is Marcus du Sautoy.

And this is our story.

You might not think it possible but the professor and I have a burning passion that we both share - the Arsenal, the greatest football team the world has ever seen.

But where we are united by football, we are divided by maths.

I'm a maths reject.

I'm quite rubbish at maths.

Even though I got a C at "O" level, it's not really my thing.

Whereas I am a maths geek.

I did my maths "O" level a year early and I got an A.

I was fourteen years old.

On a good day, I can work out the tip in a restaurant if it's 10%.

Now I'm a maths professor at Oxford University.

I also have the feeling that there is a bit more to maths than I learnt at school, or at least than I listened to at school.

I know lots of people hate maths and think it's really boring.

But I want to show Alan, show everyone in fact, that it's a wonderful, exciting subject.

Hi, Alan I'm Marcus.

Nice to meet you, all right? 'So here we go then.

'Time to get mathematical.

' On our mathematical mystery tour I want to prove to Alan that maths can help us understand everything that surrounds us.

I'm going to introduce him to some fundamental principles of maths that will expand his mind and challenge his concepts of reality.

The tasks he'll face will become harder until eventually he tackles a question so huge, it will alter the way he thinks about the universe itself.

I'm taking Alan back to his old school.

Before we embark on our odyssey I need to know exactly what he learnt and how much he remembers from his days as a maths pupil.

Oh, here we are, this is Bancroft School.

So, does it bring you out in a cold sweat? A little bit, a little bit.

I'm over the worst of the trauma.

I was the youngest boy in the school.

I'd been taken out of primary school and put here because I was quite bright.

Imagine that! And about halfway through the Sixth Form I jacked it in and went to the local college to do media studies.

Media studies, I don't know.

OK, so I've got a tough task ahead.

Right, so here's the maths classroom.

So what were you like at maths, Alan? Show us some of your maths tricks.

Well, I was, yeah right, when I was at primary school I was very good at sums, even though I say it myself.

Yeah? I was good Good at multiplication tables? I liked doing those sorts of puzzles and I knew my times tables and then when I got here I rapidly lost interest in it.

I can remember, I can remember doing Venn Diagrams.

Yeah, what's that about then? There's some things in here that are also in here and other things that are in here but aren't in there, they're in there.

Right, yeah.

I also remember two Mexicans having a tug of war.

What's the maths in that then? HE LAUGHS I used to do quite a lot of these kind of doodles.

This would be Frank Stapleton firing the ball into the top corner at White Hart Lane.

Come on back the maths, Alan! Back to the maths! My exercise book was full of that and I also remember that in the, in the maths text book, the answers were in the back, so you could cheat and then Mr Porgarides would say, "Where's your working-out?" Yeah.

"Show me your working-out.

" And you'd say, "I've thrown it away or it's at home.

" "It's kind of obvious, isn't it?" All right, OK.

But I was, I was quite good at sums, although now I'm not sure if I could do 192 times 18.

Two eights are sixteen, carry one, nine, do you do nine eights? But in a way, you know Alan, don't worry.

Honestly I wouldn't worry, you know.

I'm going to have as much trouble as you, and in some ways this is great, this is important bits of maths, but it's not the big ideas of maths, doing multiplication tables and things like that.

That's the kind of grammar of maths but it's not the Shakespeare, and that's what I want to show you.

I want to show you what the really big stories of maths are, you know.

Understand how the universe works, so that's the journey I want to take you on.

Now wipe all that out.

You don't need to know all that.

I'm going to take you and show you what maths is really about.

OK.

Now that sounds better.

You like the sound of that? Mr Porgarides never mentioned any of that.

Right, I'll wipe that out then.

Up with the Arsenal The Arsenal's our name We play at Highbury and football's our game I could tell Alan would need some convincing, so I thought I'd use football as a way inside his brain.

So without maths you wouldn't have shirt numbers, you wouldn't even have a score.

I mean Arsenal got lots and Dynamo Kiev got lots.

We don't know who won, and you see how Arsenal pass the ball around? I mean Fabregas is making all those triangles across the pitch, even quadratic equations.

Every time somebody has to work out where to stand in the box in order to volley a ball in the back of the net.

I mean, basically, these footballers are mathematical geniuses.

I have to stop you there.

I've got Ian Wright in my head, solving quadratic equations.

All right, but you know intuitively they're doing this maths all the time on the pitch.

Hmm, that's all very well for sporting legends.

How does maths apply to the mere mortals who just watch the game? Another area of maths which comes into sport a lot of course is probability, in assessing odds.

What is the odds that Arsenal are going to win? You've got loads of maths, but this is a maths that people are really bad at.

They just can't do probability.

Marcus, I've got my match day programme, and on the back of it there's the lucky number for the lucky number draw.

Exactly, to win a signed shirt or something.

What's my probability? What are my chances? How many seasons do you think you've got to come here before you've got a good chance of winning this? You know I'm always pretty optimistic.

I come to every home game, you know.

I mean I'm hoping.

I'm 42 now.

I hope in the next 20 years it'll come up.

Well, OK, let's see.

There's 40,000, you've got 19 home games a season.

When you think about it's, you're going to have to come for 2,000 seasons before there's a likelihood of you winning that.

2,000 years? 2,000 years' worth of Arsenal matches.

If you had started buying these you know, when Christ is crucified then OK you might have won once, so that is how bad you are at assessing the odds Arsene Wenger comes on the big screen and reads the numbers and I always get my programme out.

"Congratulations, you're a winner, Alan.

" But honestly, probability says you're unlikely to win tonight I'm afraid.

What's it like then for the lottery then? I don't do it, but if I did? It's horrendous.

It's like one in 14 million or something.

You'd have to This is about one in a thousand lifetimes.

Yeah exactly.

If Neanderthal man, if the first thought he'd had was to buy a lottery ticket, he might have won once by now.

That's how crazy it is to buy a lottery ticket.

So we are absolutely useless at doing probability.

You still think you're going to win, don't you? What about the first goal scorer? You will win tonight.

I want Alan to realise that having faith in maths, rather than blind optimism, will give him a clearer, more accurate understanding of the world around him.

The best way to show him this, is with a game.

Not football, but a probability game.

Inspired by an American quiz show and named after its host, the Monty Hall Game confounded even some of the world's leading mathematicians.

I expect it to challenge Alan's faith in his own common sense and show him the power of logical thinking.

So Alan, in this game show you've got the chance to win an E-type Jag.

Nice, very good.

Or if you lose you get one of these two goats.

OK? I've got enough goats.

You've got enough goats, all right.

I'm going to scramble these up so close your eyes.

OK.

Right, OK you can open them.

I want you to choose one of these three pots and try and identify where you think the car is.

Pot three.

You're going for pot three, OK.

So I'm going to show you that there is a goat under pot number one.

Right.

OK, so now we're just down to two pots.

Excellent.

I'm going to give you the chance now to change your mind if you want to.

You can move from pot three to pot two, or you can stick with your original choice.

I'll stick with three.

So why have you done that? Well, it's two pots 50/50 and if I change to two and you lift it up and it's three, that is quite annoying.

You are going to kick yourself, "Why didn't I have the courage of my convictions?" So I'll stick.

OK I think that most people's intuition is that it is 50/50 now, OK? But actually if you had changed to pot two there's the car under pot two and in fact the maths tells you that if you change your mind, actually you double your chances.

When you're left with two, it's 50/50.

I know it looks like it's 50/50 but actually if you follow through the maths, you doubled your chances.

I just think that's no.

You don't believe me? No, cos it's one or the other.

OK, let's do an experiment.

We set the game up so we could each play 20 times.

When I play I'll change my choice when offered the chance.

Whereas I will never change my first choice and I expect to win more or less the same number of cars as Marcus.

'He doesn't realise it but the maths tells me he's doomed.

'And this is why.

' There are three different scenarios.

In the first scenario I choose this cup with my first choice.

Now, remember, Alan knows where the car is so doesn't want to reveal it and will always choose to reveal the other goat.

But now if I change my pick, I'll win the car.

In the second scenario, I pick the other cup hiding an animal with my first choice.

Again Alan, who knows where the car is, will reveal the remaining animal.

I'll change and I'll win.

It's only in the third scenario when I pick the car that I lose because I'll change to an animal but in two out of the three scenarios, by changing, I win.

OK, let's put the maths to the test, so let's do the first round.

'Because Alan won't be changing his choice, he's twice as likely to end up with an animal.

' You've chosen pot number two.

I have to work out So I'm going to reveal a goat here and you're sticking with your original choice? Yes.

Yeah, OK.

So let's see what you won this time.

Ah, a sheep.

'That's a bad start admittedly, 'but there are 19 more chances to win a few more cars.

' I can see how this is going to go.

OK pot three? A cow.

You want to stick it on the pile.

Another sheep.

Another pig, another pig.

You haven't won any cars.

Now you are beginning to believe in the maths.

I'm beginning to believe they haven't put any cars in.

Piglets.

This is unbelievable.

Alan's not alone.

When this game was first devised, many respected mathematicians struggled to understand its logic.

One pig.

And another piglet.

OK.

Two cars.

Two cars and basically you've got a farm there, 18 farmyard animals.

And a couple of the Dukes of Hazard.

So are you going to try my strategy? See if I can win a few more cars than your paltry I'm on the back foot but I'm OK, so I've chosen pot number one, and so you've got to reveal an animal and then I'm going to swap.

OK, there's a little lamb.

There's a little lamb there.

I'm going to swap three, double my chances hopefully and.

.

OK, well done.

I've won a car on the first go.

Excellent, put it down there.

Another car.

Another car.

Three cars.

You've won, you've won already.

I've won already, and another car.

I'm going to give you a pig.

Yes, and another car.

Yes.

Bad luck.

And another car.

And another car.

Another cow.

Another car.

Another car.

And another car.

And another car.

Look at your cars over there.

I'm completely trouncing you.

I don't want to look at my car park.

Things haven't gone well.

And a car again.

Another car.

HE LAUGHS Right, one, two, three, four, five, six, seven, eight, nine, ten, 11, 12, 13, 14, 15, 16 cars.

Right, so do you believe in the maths now? It still seems like an exceptional result for you.

You think it's a, yeah, that is actually exceptional.

I would expect that my strategy would give me twice as many cars as animals.

OK, right.

Now I think, I think I've got this now.

Right, go on, convince me.

OK.

I'll go on three.

I reveal an animal as the host.

That one I swap.

I get a car.

Yeah.

Two out of three times.

Two out of three times, this strategy gets you the car and that's the power of maths.

This is totally counter-intuitive.

I didn't get this the first time I looked at it.

I thought surely it's 50/50 but I followed the math through.

I put on my X-Ray specs that makes me see the world in a mathematical way and suddenly I could see this pattern emerging and that gives me power you know.

I won this game.

You, you only got two I got 16.

Convinced, yet? Yes.

Absolutely.

Excellent.

Yes.

So if I put it on two 'OK, I think I get this.

'My common sense made me lose, maths made Marcus win.

'But will thinking mathematically really change my view of the entire universe?' Taking a mathematical viewpoint isn't just helpful, it's a wonderful enriching and creative way of thinking.

To help Alan realise this, I want to try and show him something that no-one has ever seen.

So we're off to a strange and foreign land, Alan, Paris, but actually somewhere ever stranger than that because these are our tickets to the fourth dimension.

I think this is going to be the real test whether I can get you to see in your mind's eye in four dimensions.

So are you ready to be sucked off into? Sorry? THEY LAUGH Paris is a lively town.

Where are we staying for God's sake? The fourth dimension is more exciting than I thought it was going to be.

It just says Gare du Nord on my ticket.

I don't know what you've bought.

Yeah, well.

If Alan was ever going to make it to the fourth dimension, first I had to mentally prepare him for the journey.

OK, Alan, let me take you into the fourth dimension.

The best place to start is with some smaller dimensions.

Let's start with one dimension.

Let's draw a line, so I'm going to draw a line and I've got two points at the end of those lines, OK? OK.

But now, so there's, let's push this into two dimensions, so I'm going to take my line and push it Width.

So I've added some width to it and now I've got a square.

OK.

OK I've moved this line and got a square and so now the square has four points on it, OK? Height and width.

Height and width.

I'm with you so far.

OK, good, so let's push this shape into the third dimension.

So depth.

OK, depth towards you.

OK, now, this square has become a cube.

Right.

And I've got four points at the base and I've pushed it, the depth and I've got four points at the other side of this cube, so how many points in total? Eight.

Exactly.

Now I'm going to take my cube and I'm going to push it into the fourth dimension.

Phssssh! OK, well we haven't got a fourth dimension to push it into, but in my mind's eye you can start tosort of extrapolate, go a little bit further.

OK? So I've got that cubed square and I pushed it into three dimensions.

That added an extra four points, so if I take this cube with its eight points and now we are going to push all the points of the cube one step into the fourth dimension, how many points does this new shape have? More Well, four Two to four to eight So, sixteen? There you go.

You spotted the pattern there.

That's what the mathematician does.

But where is it? Where is it? It's here.

OK.

It's not physical but it's, you know, we've done an act, an imaginative act.

It's like creating, sort of, a magical world in a novel or something like that.

We've I can't actually create this thing for you but we've pushed this shape with its base, the three-dimensional cube, to the four dimensional cube and now we have a four-dimensional object in our mind's eye.

And you've already explored, you've seen sixteen points then, on this four-dimensional cube.

Cool.

I think it's pretty cool.

I'm pretty pleased with myself.

To help Alan travel further into the fourth dimension, I brought him here to where they've attempted to build the impossible - a real world four-dimensional cube.

Et voila! Oh, wow! It's a four-dimensional cube, stuck here in Paris.

Cool.

But I'm not sure how many Parisians know it's a four-dimensional cube but I'm not a 100% sure I know it's a four-dimensional cube.

I mean it's not of course a real one but it's enough to explore, actually, a lot of its features.

You can't build a real four-dimensional cube.

But you can represent one in three dimensions, as they have done here.

Or you can even represent it in two dimensions.

Either way, they still retain many of the four-dimensional cube's characteristics.

You can see the sixteen points, thirty-two edges and twenty-four faces.

It's amazing this thing actually keeps all the information about this cube.

All the points are here, all the edges are here so you can explore some of these shapes in four dimensions, even in our three dimensional world.

Imagine a four-dimensional baguette with thirty-two edges, a four-dimensional croque-monsieur with sixteen points, or how about 4-D snails? They could conquer the world.

So Marcus, I'm getting it now.

I can see that someone has made up this fourth dimension, and that's quite fun.

That's a fun bit of maths.

It certainly is.

But I don't I don't get what the point of it is.

Well, partly the point is life would be very boring if you were stuck in three dimensions.

More exciting to take this imaginative leap and create something.

But is it any use? That is interesting.

When mathematicians came up with it, they weren't thinking about uses but today we use four-dimensional, 5-dimensional geometry all the time, actually.

Digital technology - for example, if you've got a picture on your computer, that's stored on your computer using very high dimensional spaces to store that information but that wasn't the motivation for creating it.

It was thisjust an idea exploring new worlds, imaginative worlds.

And this was all done about 100 years ago? The 19th century, before computers were around.

But, you know, that 19th-century That stuff lay there and now it's used? And suddenly you find yourself in a new world.

Look here we are, in a new world! I'm not sure what's going to happen now.

I know what's going to happen.

We are going to have a baguette.

Seen from the fourth dimension, maths looks very different.

It's the opposite of what I'd always imagined it to be.

It's actually wonderfully creative.

Marcus has really shown me something I had no idea existed and I didn't I thought when he was saying to me, you can create the fourth dimension, I couldn't see the point and then, when the penny dropped, that all this mathematics was done in the 19th century by people just creatively playing with numbers for fun, and excitement and then no-one really had any particular use for it until the 20th and 21st century, it dawned on me why Marcus gets so excited about the creative side of maths because it's not just arithmetic.

It's It's all about playing.

It's play, really, and it's Marcus's almost childlike enthusiasm for the potential of numbers and shapes, it's from that sense of what can happen and it's the excitement that you might as a mathematician, make discoveries and connections now and who knows where other mathematicians will take it in the future? I'm quite pleased with my progress so far.

I want to know a bit more about the world Marcus inhabits, so I have come to Oxford.

This is where Professor Du Sautoy and his chums do the hard stuff.

This is quite quite a spooky place.

It's very peaceful.

A little bit cut-off from the outside world.

I just spoke to someone from the college, I said Marcus can't be here.

She said, "Are you here in his stead?" and I said, "Kind of", and then she introduced me to a friend, "Sorry, is it Professor Davies?" I said, "No, it's not.

Not Professor.

" I should have said yeah.

It could have opened some doors.

In the corridors of the mathematical institute, you can almost smell the maths.

I hoped that by merely being here, some of it might rub off on me.

That's Marcus's room.

That's Marcus's room.

KNOCKS ON DOOR I know he's not in.

Ah, ha ha ha! Hey, look, he actually does actual sums all day.

God I really think he's been dumbing down for me.

That's a different language he speaks.

Marcus Du Sautoy, Zeta Functions of Groups and Rings.

"This book considers a new class of noncommutative Zeta functions "which encode the structure of the subgroup lattice in infinite groups.

" I'm really hoping for something like The House At Pooh Corner.

Just some sign oflife.

"Instead of taking the parallel projections of V on the axes of the above coordinate system, "we can just as well take its orthogonal projections on these axes.

" I've never been in an office before and there wasn't anything I wanted to nick.

Ah, the mathematics common room.

'These must be genuine maths students - 'the sort Marcus normally teaches.

' Do you mind if I .

No, come along.

If you have a projective variety over a finite field you can compute its zeta function which basically counts the number of points on the variety.

Are you making that up? Is that an actual equation? It's an actual equation and you want to calculate this thing, basically.

OK, so change that to X over F Q to the K x 2 to the K, over K.

So basically you stick in K= oh, and put 1 there, because you don't want to divide by 0.

Ha! THEY LAUGH Seven years.

Have you learnt nothing?! Are you allowed to write on the tables then? Oh, it's encouraged.

Do you know Do you know what this is? It's two Mexicans having a tug of war.

Oh, there we go! THEY LAUGH Very good.

I thought you might like it.

Oh, who am I kidding? Mr Porgarides was right, I'm no mathematician.

I don't think I ever will be.

I feel like no matter what I do, I don't feel like I've got the maths brain.

I don't think that's true, actually.

I think we are all programmed to do maths.

I think that we're almost hard-wired to be able to do mathematics and people say, "Oh, you've got a math brain," but I believe we've all got maths brains.

Actually we are going to go and do an experiment to see whether my thesis is actually true - to see what my brain is doing when I'm doing maths and see what your brain is doing.

And see if there is a difference.

Very little, I suspect.

I don't really think I've got the thing.

I don't think I've got the maths brain.

I think that my maths bit of brain is too small.

OK, let's prove that you're wrong.

I'm just going to mark the centre point of your head and 'I'd arranged to have our brains scanned while they're engaged in the act of mathematics.

' Sothis isn't going to hurt? You're not going to burrow a hole in my head? I think you'll be OK.

It's a completely non-invasive and safe technique.

'Alan and I would be asked the same set of maths questions 'as twenty-four detectors attached to a rather fetching beanie hat would look at how much oxygenated blood 'would flood the maths bit of our brains as we worked out the answers.

' It feels like Mastermind.

Right I should concentrate.

So, just relax and I'm just going to ask you a series of questions.

'First the really easy questions.

' 3 x 8? 5 x 6? Thirty.

7 + 8? I've got nothing to lose.

It's all on Marcus.

He's the maths professor.

He's nervous.

He went to the loo beforehand.

I heard it.

How many one-centimetre sugar cubes would fit into a cube-shaped box with four-centimetre sides? 'Come on Alan, you can do this.

'I may be no Oxford don but I know my times tables.

'Almost.

' In a class of 50 students, 18 play football, 26 play hockey, and 2 play both football and hockey.

How many students in the class don't play either football of hockey? OK that's it, Marcus.

Well done, all finished.

You're released now.

'My time has come.

' 'Oh, I'm ready.

' Alan, would you like to come through? Right.

I saw this Star Trek episode once where Spock had his brain taken out.

They just did it behind a screen and he was chatting.

That could be happening.

Well, that was extremely stressful and I probably lost my job after getting some of the really simple calculations wrong so I think I'm a bit afraid that Alan is going to totally freak out.

On the other hand, comedians are very good at lateral thinking so he might find ways round these things so it'll be intriguing to see what has sparked in his brain.

Up with the Arsenal The Arsenal's our name We play at Highbury And football's our game 9 - 4? 5.

15 - 9? 6.

3 x 9? 27.

3 + 9? 12.

A lady, when asked her age, replied that she was thirty years old not counting Saturdays and Sundays.

What was her real age? 26? If you roll a pair of dice, what is the probability that the difference between the numbers on the two dice is exactly 1? I don't know how I don't know how you say it 5:1? And rest.

OK, well done.

That's finished.

How was that? I needed more time.

Needed more time? ALAN AND MARCUS LAUGH OK? I feel like I've been cooked.

Yeah, I can see the smoke as well.

But is what's inside any different? If you want to come in, I'll show you what we've measured inside your brains.

Yes.

Whose is that really hot one? Here we go, this is you, Marcus and this is you processing the easy sums.

And is this left, right-hand side of the brain.

This is the left-hand side and this is the right-hand side.

The measurement that we were making.

What we can see here is some quite global, quite spread-out activation in your brain as you were doing those mental arithmetic sums.

If however we now look at when you were looking at processing the more difficult calculating tasks, we see a slightly more localised effect and we're not seeing effects that are spread out quite so much over the brain.

I found the easy ones more stressful cos I knew that I was If I get those wrong everyone will just I'll lose my job.

That fits the data that we see from what we measured in your brain and so Alan, this is you during your easy task, and the characterisation of you processing the easy mental arithmetic sums was that you had much more localised activation than Marcus did.

'Look at that.

I'm doing maths without panicking.

'Eat your heart out, Prof.

' If we look again to the hard sums now for Alan, and again similar area to the area that you were using for the easy sums but not quite as broad a spread.

Marcus didn't have any, why have I got blue and Marcus didn't have any blue? This is regions of potentially deactivation.

What you're saying is this is deep space? Yes.

So this bit of the brain is the bit where you know that there's gonna be maths action? That's been shown before to be associated with maths ability.

Yet I've still got 90% blue.

But there is I think you need to focus on the region of activation there.

Be positive.

Be more positive about your ability.

You're very kind but look at I mean, it's like the Pacific and that's Hawaii.

Bermuda Triangle, maybe.

You have a Bermuda Triangle.

'The blue says I am a bit of a mathematical slacker, 'but where there's red, there's hope.

' We're definitely seeing a region of your brain activating when you're processing maths.

So you do have a maths brain.

I have a maths brain.

It's just a little It's been scientifically proven.

It's small but it's there.

To give Alan's maths brain the work out it needs, I want to introduce him to the thing that first attracted me to maths - the idea that once something is shown to be true, it is true for eternity.

So proving something in maths is the closest one can come to immortality.

And how better to demonstrate this to Alan than with my favourite sort of numbers - prime numbers.

So, Marcus, I have a memory from school of prime numbers.

Yeah.

That they can't be divided up.

Exactly.

And that's all That's as far as it went? I can't remember why they taught us that You're absolutely right, a prime number is a number which you can't divide.

It's indivisible, only divisible by itself and 1.

So what? They're so important for mathematics cos they're really the building blocks of the whole of my subject.

If you take a number like 15 - OK, is that a prime? No.

No.

What's it divisible by? 3 and 5.

Exactly, and 3 and 5, they're primes - indivisible numbers, so for me, all numbers are built by multiplying prime numbers together and that's why they're important.

You get mathematics from these primes.

2,000 years ago, people wondered if these important numbers went on forever, or if there were a finite number of them.

A Greek mathematician called Euclid settled the debate with a brilliant mathematical proof and this is how he did it.

So we are going to start with somewhere where you're happy, OK? So, what was a prime number? Can you remember what the property of a prime was? It's divisible only by itself or 1.

Exactly, so they're the indivisible numbers and if it isn't prime, that means that you can divide it, divide it and divide it, until you get down to primes which divide that number, OK? So let's set off on our journey, this proof.

Now let's suppose that you think, crazy man, that there are actually only finitely many primes.

Fool that you are.

So you think there are a finite number of primes.

I think there are finitely many primes.

OK, I say, "No, no, no.

" I can prove to you why there are actually infinitely many primes.

So, let's say, give me the list of primes that you think there are.

2 2, OK? 2, 3 Yeah, 3.

5.

5.

7.

7, OK.

11.

OK, you can stop there, OK.

Suppose you think that is all the primes there are, you don't need any more.

OK, now I'm going to show you how the ancient Greeks proved there must be some primes which are missing from your list, OK? So here's the trick.

What you do is you multiply them all together.

We're going to do 2 x 3 x 5 x 7 x 11.

Which gives a grand total of 2,310, a figure that because it's been made by multiplying two, three, five, seven and 11, has to be divisible by them.

Now here is the act of genius.

This is Euclid's idea.

He says add one to this number.

OK, so let's look at this new number.

Is this number divisible by any of the primes in your list? I don't know.

OK, well, look, is it divisible by two? Well Well, no.

No, because it's an odd number.

So it's not divisible by two.

Whereas before the number was perfectly divisible by two, three, five, seven, and 11.

Now if you were to divide it by these same numbers because you've added one, you would always have one left over.

So I've built this number such that this number is not divisible by two, three, five, seven, or 11, because you always get remainder one.

So it's only divisible so is it a new prime number? Well, there you go, so this is either a new prime number or There may be a number that divides it.

It's got to be a prime which divides it.

But it's not one of those.

So actually you missed some.

So there is another prime out there.

Yeah, you missed one.

Brilliant.

That is an act of genius, Alan.

Is it? Absolutely.

Because that is I must be learning something.

You got to the heart of the problem.

It doesn't matter how many primes you multiply together.

By adding one you'll always discover either a new prime number or a number divisible by a prime that's not on your list.

So with this beautiful argument, Euclid's shown why the primes go on forever but he hasn't helped you to find them.

But I think what's so amazing about this, Euclid proved this 2,000 years ago, and it's as true today when we're sitting on this beach here as it was when he proved it on the beach in Alexandria.

There's something about the permanence of proof that gives you a 100% certainty, and that proof will be there forever.

I mean, this beach will disappear, crumble into sand, but that proof is indestructible.

It sort of gives you a little bit of immortality.

Euclid's name will somehow live forever because of this proof.

It's the beauty of mathematics, this kind of power of proof.

Well, clever Euclid.

He was a clever guy.

I feel pretty confident with prime numbers now.

Yeah.

It does make you want to know more, that's for sure.

The more he talks to you, the more you want to go, "And then what?" It's very enthusiastic, it's passionate, it's amazing.

I don't know where that comes from in a man.

But I much prefer it to the downtrodden apathy of the rest of you.

Prime numbers don't just allow us a glimpse of the infinite.

Recently we've discovered a strange new connection between primes and fundamental properties of the natural world.

To understand this connection, Alan will need to get to grips with some really challenging mathematics.

He's been coping well so far, but what we're about to do is several degrees harder.

OK.

If it's anything to do with primes, I'm ready.

In the 19th century, mathematicians looked at the apparently random distribution of the prime numbers they'd discovered and realised there was some order to them.

A German mathematician called Bernhard Riemann then came up with an exceedingly complicated equation which allowed him to predict the distribution of the primes using this strange three-dimensional graph.

The contours of Riemann's graph appear to expose the hidden pattern of the primes that eluded mathematicians for centuries.

But a recent breakthrough in analysing this graph has unearthed some startling parallels between the abstract world of the primes and the physical world that surrounds us.

Which is why Marcus has brought me here to the National Physical Laboratory.

We're going to undertake an experiment.

Well, Alan, what we are going to do here is we're going to strike this quartz sphere with a ball-bearing ball.

That doesn't sound very technical.

It's not very technical.

I like this kind of physics.

But it is effective! Underneath there are contact microphones, and they convert the vibrations into an electrical signal.

OK, good.

This sounds like quite a straightforward experiment.

I guess it is, but I guess the consequences are more important than that.

This thing is vibrating It'd better be, otherwise you're wasting this man's time.

I feel a lot of pressure.

This is more than darts! I've got to hit that flat bit.

Try your best, Alan.

I've got to get I'll get quite close to it, to make sure.

Good, wow! Now look at that.

Look what I did! Look what you've done.

I threw it quite hard there.

You did! It almost went off the scale.

Sorry.

But every little peak here corresponds to a vibration of this quartz sphere, and the crucial thing you notice is that the way in which those are distributed is related to the series of prime numbers.

This means that if you analyse the distribution of the quartz's vibrations and do the same for the contours of Riemann's graph, you'll see a remarkably similar result.

So similar it cannot be a coincidence.

So some naturally occurring thing in nature that links us PHONE RINGS Is that Riemann? If only I'd known about the quartz! Is there a dial a friend? I'd like to dial a friend.

I did all those sums for years, and I just had to throw a ball-bearing at a bit of quartz! But it's in nature, it's naturally occurring and yet it coincides with the work of this genius mathematician from 200 years ago.

And it's similar enough for you to think, are we on to something? Exactly, similar enough that this can't be a coincidence.

A number pattern's so similar to naturally occurring phenomena, if you like, that maybe we are onto something.

Yeah.

I think this is this is not just a coincidence.

It's a message telling us that the maths which is going on behind this, the way this quartz is vibrating, will help us to explain the way the primes are distributed.

That odd little list of weird looking numbers that don't appear to relate to one another, it's not very pleasant to look at because we like things to go "two, four, six, eight" is actually This is where the pattern is.

They're found everywhere, and they could be the key to everything.

THEY LAUGH This mysterious connection is found elsewhere too.

On a microscopic scale, quantum energy levels in atoms like uranium match this pattern.

Bizarrely, a parallel also exists at the municipal level, in the distribution of bus arrival times in a little-known Mexican city.

There's even a connection between Riemann's graph and the distribution of parked cars in modern-day London.

I think I've converted, I've joined the maths club.

I'm ready now to go forth and do battle with the greatest maths problems of the 21st century.

If only I knew what they were! So, Marcus, what are the big maths questions that we can go after? You want to go for really big iconic problems? Yes! You can't do better than one of the seven millennium problems.

These are the really big questions of maths.

You can win a million dollars for solving any one of these problems.

That's how important they are.

A million dollars for solving a maths equation? Well, slightly more than just a maths equation, but a million dollars.

How does that sound? I'm up for it.

I can see, you're incentivised now! Cos a million divided by two is half a million.

Half a million.

I can see your maths brain is working.

Who said we were going 50/50? OK, ladies and gentlemen, step right up, just a few simple questions.

The Riemann Hypothesis, Navier-Stokes Equations, P v NP.

Try your hand at the Yang-Mills theory.

The Birch and Swinnerton-Dyer, the Hodge, or the Poincare Conjecture.

Don't be shy.

Give it a try! So Marcus, has anyone ever won the million dollars? Well, there's a guy who might be on the verge of winning one.

Grigory Perelman might have solved one of the problems that people have been thinking about for millennia, namely, what is the possible shape of the universe? But if he gets offered the million, it looks like he might actually turn it down.

He's been offered some prizes for solving this problem, and he's turned those down.

So why is he going to turn it down? Well, he's more interested in the buzz of solving maths problems than winning a million dollars.

You can spend a million dollars very quickly but that His name will live forever for solving this iconic problem called the Poincare Conjecture.

That's good for him.

As an amateur mathematician I'm in it for the prizes.

Here you are.

Oh, dear, let's see whether I can That's five with one! He's done it again.

I am a bit of a maths novice, admittedly.

I'm going to see if I can add my pennies-worth to this million-dollar shape of the universe question.

This has to be the biggest problem I've yet tackled.

Trying to imagine the shape of the universe is a problem that mathematicians have been battling with for thousands of years.

So what's the shape? I don't really see a shape there.

I just feel it just goes on and on and on and on and on and on and on.

Well, that's a model, you know.

It's infinitely big.

Stretches out to infinity, I think that's one most people think about.

Does it have an edge, a hard edge? One of the ancient ones was that it was contained in a shell and you'd hit some sort of wall.

Shaped a bit like Russell Grant? Well, maybe.

I bet it is.

But mathematicians have come up with a third idea, which I think is much more satisfying to explain what the shape of the universe is.

To explain this to Alan, I was going to take him back in time.

Well, at least to the 1970s.

If this counts as maths, I should have a PhD.

I spent my childhood playing this Asteroids game.

So this little universe that our spaceman is flying around in will be really useful for trying to understand what the shape of our universe is.

You could say it's just a rectangle.

But it's more interesting than that, because the top of the screen, when your spaceship shoots off the top, it doesn't bounce back again.

It comes back in the bottom.

So actually the bottom of the screen is joined to the top of the screen.

So what shape am I making by joining the top and the bottom of the screen up? A kind of cylinder.

Yeah, exactly.

The screen is wrapped up round into a cylinder.

So the two-dimensional universe gets wrapped round in three dimensions into a cylinder, but not only that What about when I go off stage left, where do you come? You come in stage right.

OK.

So actually the two sides, two ends of the cylinder are also joined.

So if I join these two ends of the cylinder, what shape do I get? OK, now And there's a hint in your brown paper bag.

What I brung from the pier.

Exactly.

It may be relevant.

It is.

The shape of this universe is actually a doughnut.

OK.

So the Asteroids spaceship flies around a two-dimensional universe wrapped into a three-dimensional doughnut shape.

But we live in a three-dimensional universe.

We can move left, right, up, down, and in and out.

So, you know, what's the shape of our universe? Well, this cinema could be a model of our universe.

It's a finite universe, but it could be a universe without any boundaries.

So maybe when I walk this way, and then I hit this wall here, instead of hitting it, maybe I go through and reappear on this side of the universe.

If I go through the screen, maybe I don't hit the screen but I reappear at the back of the cinema.

I've joined up the sides of the cinema, the back and the front and I've got a third dimension.

I could shoot out the top and reappear through the floor.

So this finite universe could have no boundaries just like the Atari game.

But then what's the shape of our little universe that we've created in the cinema? I could start trying to join this cinema up.

We could use this wall here.

We've got to join with the opposite wall, so we could wrap that round, to start making our large doughnut shape, but then I'm going to get stuck.

How do I join the screen with the back of the cinema, and then there's the ceiling with the floor? You're going to need another dimension.

Absolutely.

A bigger dimension.

A fourth dimension.

Just like we learnt In Paris.

In Paris.

So if you give me a fourth dimension, then I've got flexibility to join this shape up, and our little mini-cinema universe would become a doughnut in a four-dimensional world.

Are we living in a four-dimensional doughnut? We're living on the surface of that four-dimensional doughnut.

Can you bake such a thing? This is good.

For the first time in my life, I've been thinking real mathematical thoughts, not just boring old sums.

Everything I've learned so far is being brought to bear on this question of the universe.

I've put aside common sense in favour of logic.

I'm thinking about the higher dimensions that contain our own, the nature of infinity, and the importance of proof.

And I've seen the presence of maths in places I never had before.

All in all, it's got me pondering my own questions on the shape of space.

If this new model of the universe means that we could go into space, keep going and eventually end up right back where we started, does it have another even weirder implication? So does the same thing apply to light? Because if light leaves the Earth and goes out all the way across the universe, eventually it's going to come back and hit us in the back of the head.

Absolutely.

Which means that if we look up, we should be able to see ourselves.

Absolutely.

In fact one of those stars out there might be a version, a young version of our solar system, and astronomers are looking now to see whether one of those stars might in fact be our sun.

You've sold me, Marcus.

Yeah? Yeah, you've changed my view of the universe.

That's fantastic.

So is this it now? Am I at the end of my journey? Alan, this is just the beginning.

Are you angling for a series? I've got it! I can see Milton Keynes!

Me, Alan Davies, discussing the mathematics of light travelling through the universe on a hilltop in the middle of winter with an Oxford University professor? You have completely transformed how I see the universe, Marcus.

Yeah?! But I've just had a peculiar adventure.

It took me to the fourth dimension.

That looks like a fun bit of maths.

To the outer reaches of space.

Se we're living in a four-dimensional doughnut.

To the deepest recesses of my mind.

Look at it.

It's like the Pacific.

Bermuda Triangle maybe.

Yeah.

And to Weymouth.

Brilliant.

That is an act of genius, Alan.

To see if I can be convinced that maths is not the boring subject I hated at school, I've agreed to undertake a mathematical assault course, with this man as my guide.

Yes! It's 13, get in there! Prime number I told you.

His name is Marcus du Sautoy.

And this is our story.

You might not think it possible but the professor and I have a burning passion that we both share - the Arsenal, the greatest football team the world has ever seen.

But where we are united by football, we are divided by maths.

I'm a maths reject.

I'm quite rubbish at maths.

Even though I got a C at "O" level, it's not really my thing.

Whereas I am a maths geek.

I did my maths "O" level a year early and I got an A.

I was fourteen years old.

On a good day, I can work out the tip in a restaurant if it's 10%.

Now I'm a maths professor at Oxford University.

I also have the feeling that there is a bit more to maths than I learnt at school, or at least than I listened to at school.

I know lots of people hate maths and think it's really boring.

But I want to show Alan, show everyone in fact, that it's a wonderful, exciting subject.

Hi, Alan I'm Marcus.

Nice to meet you, all right? 'So here we go then.

'Time to get mathematical.

' On our mathematical mystery tour I want to prove to Alan that maths can help us understand everything that surrounds us.

I'm going to introduce him to some fundamental principles of maths that will expand his mind and challenge his concepts of reality.

The tasks he'll face will become harder until eventually he tackles a question so huge, it will alter the way he thinks about the universe itself.

I'm taking Alan back to his old school.

Before we embark on our odyssey I need to know exactly what he learnt and how much he remembers from his days as a maths pupil.

Oh, here we are, this is Bancroft School.

So, does it bring you out in a cold sweat? A little bit, a little bit.

I'm over the worst of the trauma.

I was the youngest boy in the school.

I'd been taken out of primary school and put here because I was quite bright.

Imagine that! And about halfway through the Sixth Form I jacked it in and went to the local college to do media studies.

Media studies, I don't know.

OK, so I've got a tough task ahead.

Right, so here's the maths classroom.

So what were you like at maths, Alan? Show us some of your maths tricks.

Well, I was, yeah right, when I was at primary school I was very good at sums, even though I say it myself.

Yeah? I was good Good at multiplication tables? I liked doing those sorts of puzzles and I knew my times tables and then when I got here I rapidly lost interest in it.

I can remember, I can remember doing Venn Diagrams.

Yeah, what's that about then? There's some things in here that are also in here and other things that are in here but aren't in there, they're in there.

Right, yeah.

I also remember two Mexicans having a tug of war.

What's the maths in that then? HE LAUGHS I used to do quite a lot of these kind of doodles.

This would be Frank Stapleton firing the ball into the top corner at White Hart Lane.

Come on back the maths, Alan! Back to the maths! My exercise book was full of that and I also remember that in the, in the maths text book, the answers were in the back, so you could cheat and then Mr Porgarides would say, "Where's your working-out?" Yeah.

"Show me your working-out.

" And you'd say, "I've thrown it away or it's at home.

" "It's kind of obvious, isn't it?" All right, OK.

But I was, I was quite good at sums, although now I'm not sure if I could do 192 times 18.

Two eights are sixteen, carry one, nine, do you do nine eights? But in a way, you know Alan, don't worry.

Honestly I wouldn't worry, you know.

I'm going to have as much trouble as you, and in some ways this is great, this is important bits of maths, but it's not the big ideas of maths, doing multiplication tables and things like that.

That's the kind of grammar of maths but it's not the Shakespeare, and that's what I want to show you.

I want to show you what the really big stories of maths are, you know.

Understand how the universe works, so that's the journey I want to take you on.

Now wipe all that out.

You don't need to know all that.

I'm going to take you and show you what maths is really about.

OK.

Now that sounds better.

You like the sound of that? Mr Porgarides never mentioned any of that.

Right, I'll wipe that out then.

Up with the Arsenal The Arsenal's our name We play at Highbury and football's our game I could tell Alan would need some convincing, so I thought I'd use football as a way inside his brain.

So without maths you wouldn't have shirt numbers, you wouldn't even have a score.

I mean Arsenal got lots and Dynamo Kiev got lots.

We don't know who won, and you see how Arsenal pass the ball around? I mean Fabregas is making all those triangles across the pitch, even quadratic equations.

Every time somebody has to work out where to stand in the box in order to volley a ball in the back of the net.

I mean, basically, these footballers are mathematical geniuses.

I have to stop you there.

I've got Ian Wright in my head, solving quadratic equations.

All right, but you know intuitively they're doing this maths all the time on the pitch.

Hmm, that's all very well for sporting legends.

How does maths apply to the mere mortals who just watch the game? Another area of maths which comes into sport a lot of course is probability, in assessing odds.

What is the odds that Arsenal are going to win? You've got loads of maths, but this is a maths that people are really bad at.

They just can't do probability.

Marcus, I've got my match day programme, and on the back of it there's the lucky number for the lucky number draw.

Exactly, to win a signed shirt or something.

What's my probability? What are my chances? How many seasons do you think you've got to come here before you've got a good chance of winning this? You know I'm always pretty optimistic.

I come to every home game, you know.

I mean I'm hoping.

I'm 42 now.

I hope in the next 20 years it'll come up.

Well, OK, let's see.

There's 40,000, you've got 19 home games a season.

When you think about it's, you're going to have to come for 2,000 seasons before there's a likelihood of you winning that.

2,000 years? 2,000 years' worth of Arsenal matches.

If you had started buying these you know, when Christ is crucified then OK you might have won once, so that is how bad you are at assessing the odds Arsene Wenger comes on the big screen and reads the numbers and I always get my programme out.

"Congratulations, you're a winner, Alan.

" But honestly, probability says you're unlikely to win tonight I'm afraid.

What's it like then for the lottery then? I don't do it, but if I did? It's horrendous.

It's like one in 14 million or something.

You'd have to This is about one in a thousand lifetimes.

Yeah exactly.

If Neanderthal man, if the first thought he'd had was to buy a lottery ticket, he might have won once by now.

That's how crazy it is to buy a lottery ticket.

So we are absolutely useless at doing probability.

You still think you're going to win, don't you? What about the first goal scorer? You will win tonight.

I want Alan to realise that having faith in maths, rather than blind optimism, will give him a clearer, more accurate understanding of the world around him.

The best way to show him this, is with a game.

Not football, but a probability game.

Inspired by an American quiz show and named after its host, the Monty Hall Game confounded even some of the world's leading mathematicians.

I expect it to challenge Alan's faith in his own common sense and show him the power of logical thinking.

So Alan, in this game show you've got the chance to win an E-type Jag.

Nice, very good.

Or if you lose you get one of these two goats.

OK? I've got enough goats.

You've got enough goats, all right.

I'm going to scramble these up so close your eyes.

OK.

Right, OK you can open them.

I want you to choose one of these three pots and try and identify where you think the car is.

Pot three.

You're going for pot three, OK.

So I'm going to show you that there is a goat under pot number one.

Right.

OK, so now we're just down to two pots.

Excellent.

I'm going to give you the chance now to change your mind if you want to.

You can move from pot three to pot two, or you can stick with your original choice.

I'll stick with three.

So why have you done that? Well, it's two pots 50/50 and if I change to two and you lift it up and it's three, that is quite annoying.

You are going to kick yourself, "Why didn't I have the courage of my convictions?" So I'll stick.

OK I think that most people's intuition is that it is 50/50 now, OK? But actually if you had changed to pot two there's the car under pot two and in fact the maths tells you that if you change your mind, actually you double your chances.

When you're left with two, it's 50/50.

I know it looks like it's 50/50 but actually if you follow through the maths, you doubled your chances.

I just think that's no.

You don't believe me? No, cos it's one or the other.

OK, let's do an experiment.

We set the game up so we could each play 20 times.

When I play I'll change my choice when offered the chance.

Whereas I will never change my first choice and I expect to win more or less the same number of cars as Marcus.

'He doesn't realise it but the maths tells me he's doomed.

'And this is why.

' There are three different scenarios.

In the first scenario I choose this cup with my first choice.

Now, remember, Alan knows where the car is so doesn't want to reveal it and will always choose to reveal the other goat.

But now if I change my pick, I'll win the car.

In the second scenario, I pick the other cup hiding an animal with my first choice.

Again Alan, who knows where the car is, will reveal the remaining animal.

I'll change and I'll win.

It's only in the third scenario when I pick the car that I lose because I'll change to an animal but in two out of the three scenarios, by changing, I win.

OK, let's put the maths to the test, so let's do the first round.

'Because Alan won't be changing his choice, he's twice as likely to end up with an animal.

' You've chosen pot number two.

I have to work out So I'm going to reveal a goat here and you're sticking with your original choice? Yes.

Yeah, OK.

So let's see what you won this time.

Ah, a sheep.

'That's a bad start admittedly, 'but there are 19 more chances to win a few more cars.

' I can see how this is going to go.

OK pot three? A cow.

You want to stick it on the pile.

Another sheep.

Another pig, another pig.

You haven't won any cars.

Now you are beginning to believe in the maths.

I'm beginning to believe they haven't put any cars in.

Piglets.

This is unbelievable.

Alan's not alone.

When this game was first devised, many respected mathematicians struggled to understand its logic.

One pig.

And another piglet.

OK.

Two cars.

Two cars and basically you've got a farm there, 18 farmyard animals.

And a couple of the Dukes of Hazard.

So are you going to try my strategy? See if I can win a few more cars than your paltry I'm on the back foot but I'm OK, so I've chosen pot number one, and so you've got to reveal an animal and then I'm going to swap.

OK, there's a little lamb.

There's a little lamb there.

I'm going to swap three, double my chances hopefully and.

.

OK, well done.

I've won a car on the first go.

Excellent, put it down there.

Another car.

Another car.

Three cars.

You've won, you've won already.

I've won already, and another car.

I'm going to give you a pig.

Yes, and another car.

Yes.

Bad luck.

And another car.

And another car.

Another cow.

Another car.

Another car.

And another car.

And another car.

Look at your cars over there.

I'm completely trouncing you.

I don't want to look at my car park.

Things haven't gone well.

And a car again.

Another car.

HE LAUGHS Right, one, two, three, four, five, six, seven, eight, nine, ten, 11, 12, 13, 14, 15, 16 cars.

Right, so do you believe in the maths now? It still seems like an exceptional result for you.

You think it's a, yeah, that is actually exceptional.

I would expect that my strategy would give me twice as many cars as animals.

OK, right.

Now I think, I think I've got this now.

Right, go on, convince me.

OK.

I'll go on three.

I reveal an animal as the host.

That one I swap.

I get a car.

Yeah.

Two out of three times.

Two out of three times, this strategy gets you the car and that's the power of maths.

This is totally counter-intuitive.

I didn't get this the first time I looked at it.

I thought surely it's 50/50 but I followed the math through.

I put on my X-Ray specs that makes me see the world in a mathematical way and suddenly I could see this pattern emerging and that gives me power you know.

I won this game.

You, you only got two I got 16.

Convinced, yet? Yes.

Absolutely.

Excellent.

Yes.

So if I put it on two 'OK, I think I get this.

'My common sense made me lose, maths made Marcus win.

'But will thinking mathematically really change my view of the entire universe?' Taking a mathematical viewpoint isn't just helpful, it's a wonderful enriching and creative way of thinking.

To help Alan realise this, I want to try and show him something that no-one has ever seen.

So we're off to a strange and foreign land, Alan, Paris, but actually somewhere ever stranger than that because these are our tickets to the fourth dimension.

I think this is going to be the real test whether I can get you to see in your mind's eye in four dimensions.

So are you ready to be sucked off into? Sorry? THEY LAUGH Paris is a lively town.

Where are we staying for God's sake? The fourth dimension is more exciting than I thought it was going to be.

It just says Gare du Nord on my ticket.

I don't know what you've bought.

Yeah, well.

If Alan was ever going to make it to the fourth dimension, first I had to mentally prepare him for the journey.

OK, Alan, let me take you into the fourth dimension.

The best place to start is with some smaller dimensions.

Let's start with one dimension.

Let's draw a line, so I'm going to draw a line and I've got two points at the end of those lines, OK? OK.

But now, so there's, let's push this into two dimensions, so I'm going to take my line and push it Width.

So I've added some width to it and now I've got a square.

OK.

OK I've moved this line and got a square and so now the square has four points on it, OK? Height and width.

Height and width.

I'm with you so far.

OK, good, so let's push this shape into the third dimension.

So depth.

OK, depth towards you.

OK, now, this square has become a cube.

Right.

And I've got four points at the base and I've pushed it, the depth and I've got four points at the other side of this cube, so how many points in total? Eight.

Exactly.

Now I'm going to take my cube and I'm going to push it into the fourth dimension.

Phssssh! OK, well we haven't got a fourth dimension to push it into, but in my mind's eye you can start tosort of extrapolate, go a little bit further.

OK? So I've got that cubed square and I pushed it into three dimensions.

That added an extra four points, so if I take this cube with its eight points and now we are going to push all the points of the cube one step into the fourth dimension, how many points does this new shape have? More Well, four Two to four to eight So, sixteen? There you go.

You spotted the pattern there.

That's what the mathematician does.

But where is it? Where is it? It's here.

OK.

It's not physical but it's, you know, we've done an act, an imaginative act.

It's like creating, sort of, a magical world in a novel or something like that.

We've I can't actually create this thing for you but we've pushed this shape with its base, the three-dimensional cube, to the four dimensional cube and now we have a four-dimensional object in our mind's eye.

And you've already explored, you've seen sixteen points then, on this four-dimensional cube.

Cool.

I think it's pretty cool.

I'm pretty pleased with myself.

To help Alan travel further into the fourth dimension, I brought him here to where they've attempted to build the impossible - a real world four-dimensional cube.

Et voila! Oh, wow! It's a four-dimensional cube, stuck here in Paris.

Cool.

But I'm not sure how many Parisians know it's a four-dimensional cube but I'm not a 100% sure I know it's a four-dimensional cube.

I mean it's not of course a real one but it's enough to explore, actually, a lot of its features.

You can't build a real four-dimensional cube.

But you can represent one in three dimensions, as they have done here.

Or you can even represent it in two dimensions.

Either way, they still retain many of the four-dimensional cube's characteristics.

You can see the sixteen points, thirty-two edges and twenty-four faces.

It's amazing this thing actually keeps all the information about this cube.

All the points are here, all the edges are here so you can explore some of these shapes in four dimensions, even in our three dimensional world.

Imagine a four-dimensional baguette with thirty-two edges, a four-dimensional croque-monsieur with sixteen points, or how about 4-D snails? They could conquer the world.

So Marcus, I'm getting it now.

I can see that someone has made up this fourth dimension, and that's quite fun.

That's a fun bit of maths.

It certainly is.

But I don't I don't get what the point of it is.

Well, partly the point is life would be very boring if you were stuck in three dimensions.

More exciting to take this imaginative leap and create something.

But is it any use? That is interesting.

When mathematicians came up with it, they weren't thinking about uses but today we use four-dimensional, 5-dimensional geometry all the time, actually.

Digital technology - for example, if you've got a picture on your computer, that's stored on your computer using very high dimensional spaces to store that information but that wasn't the motivation for creating it.

It was thisjust an idea exploring new worlds, imaginative worlds.

And this was all done about 100 years ago? The 19th century, before computers were around.

But, you know, that 19th-century That stuff lay there and now it's used? And suddenly you find yourself in a new world.

Look here we are, in a new world! I'm not sure what's going to happen now.

I know what's going to happen.

We are going to have a baguette.

Seen from the fourth dimension, maths looks very different.

It's the opposite of what I'd always imagined it to be.

It's actually wonderfully creative.

Marcus has really shown me something I had no idea existed and I didn't I thought when he was saying to me, you can create the fourth dimension, I couldn't see the point and then, when the penny dropped, that all this mathematics was done in the 19th century by people just creatively playing with numbers for fun, and excitement and then no-one really had any particular use for it until the 20th and 21st century, it dawned on me why Marcus gets so excited about the creative side of maths because it's not just arithmetic.

It's It's all about playing.

It's play, really, and it's Marcus's almost childlike enthusiasm for the potential of numbers and shapes, it's from that sense of what can happen and it's the excitement that you might as a mathematician, make discoveries and connections now and who knows where other mathematicians will take it in the future? I'm quite pleased with my progress so far.

I want to know a bit more about the world Marcus inhabits, so I have come to Oxford.

This is where Professor Du Sautoy and his chums do the hard stuff.

This is quite quite a spooky place.

It's very peaceful.

A little bit cut-off from the outside world.

I just spoke to someone from the college, I said Marcus can't be here.

She said, "Are you here in his stead?" and I said, "Kind of", and then she introduced me to a friend, "Sorry, is it Professor Davies?" I said, "No, it's not.

Not Professor.

" I should have said yeah.

It could have opened some doors.

In the corridors of the mathematical institute, you can almost smell the maths.

I hoped that by merely being here, some of it might rub off on me.

That's Marcus's room.

That's Marcus's room.

KNOCKS ON DOOR I know he's not in.

Ah, ha ha ha! Hey, look, he actually does actual sums all day.

God I really think he's been dumbing down for me.

That's a different language he speaks.

Marcus Du Sautoy, Zeta Functions of Groups and Rings.

"This book considers a new class of noncommutative Zeta functions "which encode the structure of the subgroup lattice in infinite groups.

" I'm really hoping for something like The House At Pooh Corner.

Just some sign oflife.

"Instead of taking the parallel projections of V on the axes of the above coordinate system, "we can just as well take its orthogonal projections on these axes.

" I've never been in an office before and there wasn't anything I wanted to nick.

Ah, the mathematics common room.

'These must be genuine maths students - 'the sort Marcus normally teaches.

' Do you mind if I .

No, come along.

If you have a projective variety over a finite field you can compute its zeta function which basically counts the number of points on the variety.

Are you making that up? Is that an actual equation? It's an actual equation and you want to calculate this thing, basically.

OK, so change that to X over F Q to the K x 2 to the K, over K.

So basically you stick in K= oh, and put 1 there, because you don't want to divide by 0.

Ha! THEY LAUGH Seven years.

Have you learnt nothing?! Are you allowed to write on the tables then? Oh, it's encouraged.

Do you know Do you know what this is? It's two Mexicans having a tug of war.

Oh, there we go! THEY LAUGH Very good.

I thought you might like it.

Oh, who am I kidding? Mr Porgarides was right, I'm no mathematician.

I don't think I ever will be.

I feel like no matter what I do, I don't feel like I've got the maths brain.

I don't think that's true, actually.

I think we are all programmed to do maths.

I think that we're almost hard-wired to be able to do mathematics and people say, "Oh, you've got a math brain," but I believe we've all got maths brains.

Actually we are going to go and do an experiment to see whether my thesis is actually true - to see what my brain is doing when I'm doing maths and see what your brain is doing.

And see if there is a difference.

Very little, I suspect.

I don't really think I've got the thing.

I don't think I've got the maths brain.

I think that my maths bit of brain is too small.

OK, let's prove that you're wrong.

I'm just going to mark the centre point of your head and 'I'd arranged to have our brains scanned while they're engaged in the act of mathematics.

' Sothis isn't going to hurt? You're not going to burrow a hole in my head? I think you'll be OK.

It's a completely non-invasive and safe technique.

'Alan and I would be asked the same set of maths questions 'as twenty-four detectors attached to a rather fetching beanie hat would look at how much oxygenated blood 'would flood the maths bit of our brains as we worked out the answers.

' It feels like Mastermind.

Right I should concentrate.

So, just relax and I'm just going to ask you a series of questions.

'First the really easy questions.

' 3 x 8? 5 x 6? Thirty.

7 + 8? I've got nothing to lose.

It's all on Marcus.

He's the maths professor.

He's nervous.

He went to the loo beforehand.

I heard it.

How many one-centimetre sugar cubes would fit into a cube-shaped box with four-centimetre sides? 'Come on Alan, you can do this.

'I may be no Oxford don but I know my times tables.

'Almost.

' In a class of 50 students, 18 play football, 26 play hockey, and 2 play both football and hockey.

How many students in the class don't play either football of hockey? OK that's it, Marcus.

Well done, all finished.

You're released now.

'My time has come.

' 'Oh, I'm ready.

' Alan, would you like to come through? Right.

I saw this Star Trek episode once where Spock had his brain taken out.

They just did it behind a screen and he was chatting.

That could be happening.

Well, that was extremely stressful and I probably lost my job after getting some of the really simple calculations wrong so I think I'm a bit afraid that Alan is going to totally freak out.

On the other hand, comedians are very good at lateral thinking so he might find ways round these things so it'll be intriguing to see what has sparked in his brain.

Up with the Arsenal The Arsenal's our name We play at Highbury And football's our game 9 - 4? 5.

15 - 9? 6.

3 x 9? 27.

3 + 9? 12.

A lady, when asked her age, replied that she was thirty years old not counting Saturdays and Sundays.

What was her real age? 26? If you roll a pair of dice, what is the probability that the difference between the numbers on the two dice is exactly 1? I don't know how I don't know how you say it 5:1? And rest.

OK, well done.

That's finished.

How was that? I needed more time.

Needed more time? ALAN AND MARCUS LAUGH OK? I feel like I've been cooked.

Yeah, I can see the smoke as well.

But is what's inside any different? If you want to come in, I'll show you what we've measured inside your brains.

Yes.

Whose is that really hot one? Here we go, this is you, Marcus and this is you processing the easy sums.

And is this left, right-hand side of the brain.

This is the left-hand side and this is the right-hand side.

The measurement that we were making.

What we can see here is some quite global, quite spread-out activation in your brain as you were doing those mental arithmetic sums.

If however we now look at when you were looking at processing the more difficult calculating tasks, we see a slightly more localised effect and we're not seeing effects that are spread out quite so much over the brain.

I found the easy ones more stressful cos I knew that I was If I get those wrong everyone will just I'll lose my job.

That fits the data that we see from what we measured in your brain and so Alan, this is you during your easy task, and the characterisation of you processing the easy mental arithmetic sums was that you had much more localised activation than Marcus did.

'Look at that.

I'm doing maths without panicking.

'Eat your heart out, Prof.

' If we look again to the hard sums now for Alan, and again similar area to the area that you were using for the easy sums but not quite as broad a spread.

Marcus didn't have any, why have I got blue and Marcus didn't have any blue? This is regions of potentially deactivation.

What you're saying is this is deep space? Yes.

So this bit of the brain is the bit where you know that there's gonna be maths action? That's been shown before to be associated with maths ability.

Yet I've still got 90% blue.

But there is I think you need to focus on the region of activation there.

Be positive.

Be more positive about your ability.

You're very kind but look at I mean, it's like the Pacific and that's Hawaii.

Bermuda Triangle, maybe.

You have a Bermuda Triangle.

'The blue says I am a bit of a mathematical slacker, 'but where there's red, there's hope.

' We're definitely seeing a region of your brain activating when you're processing maths.

So you do have a maths brain.

I have a maths brain.

It's just a little It's been scientifically proven.

It's small but it's there.

To give Alan's maths brain the work out it needs, I want to introduce him to the thing that first attracted me to maths - the idea that once something is shown to be true, it is true for eternity.

So proving something in maths is the closest one can come to immortality.

And how better to demonstrate this to Alan than with my favourite sort of numbers - prime numbers.

So, Marcus, I have a memory from school of prime numbers.

Yeah.

That they can't be divided up.

Exactly.

And that's all That's as far as it went? I can't remember why they taught us that You're absolutely right, a prime number is a number which you can't divide.

It's indivisible, only divisible by itself and 1.

So what? They're so important for mathematics cos they're really the building blocks of the whole of my subject.

If you take a number like 15 - OK, is that a prime? No.

No.

What's it divisible by? 3 and 5.

Exactly, and 3 and 5, they're primes - indivisible numbers, so for me, all numbers are built by multiplying prime numbers together and that's why they're important.

You get mathematics from these primes.

2,000 years ago, people wondered if these important numbers went on forever, or if there were a finite number of them.

A Greek mathematician called Euclid settled the debate with a brilliant mathematical proof and this is how he did it.

So we are going to start with somewhere where you're happy, OK? So, what was a prime number? Can you remember what the property of a prime was? It's divisible only by itself or 1.

Exactly, so they're the indivisible numbers and if it isn't prime, that means that you can divide it, divide it and divide it, until you get down to primes which divide that number, OK? So let's set off on our journey, this proof.

Now let's suppose that you think, crazy man, that there are actually only finitely many primes.

Fool that you are.

So you think there are a finite number of primes.

I think there are finitely many primes.

OK, I say, "No, no, no.

" I can prove to you why there are actually infinitely many primes.

So, let's say, give me the list of primes that you think there are.

2 2, OK? 2, 3 Yeah, 3.

5.

5.

7.

7, OK.

11.

OK, you can stop there, OK.

Suppose you think that is all the primes there are, you don't need any more.

OK, now I'm going to show you how the ancient Greeks proved there must be some primes which are missing from your list, OK? So here's the trick.

What you do is you multiply them all together.

We're going to do 2 x 3 x 5 x 7 x 11.

Which gives a grand total of 2,310, a figure that because it's been made by multiplying two, three, five, seven and 11, has to be divisible by them.

Now here is the act of genius.

This is Euclid's idea.

He says add one to this number.

OK, so let's look at this new number.

Is this number divisible by any of the primes in your list? I don't know.

OK, well, look, is it divisible by two? Well Well, no.

No, because it's an odd number.

So it's not divisible by two.

Whereas before the number was perfectly divisible by two, three, five, seven, and 11.

Now if you were to divide it by these same numbers because you've added one, you would always have one left over.

So I've built this number such that this number is not divisible by two, three, five, seven, or 11, because you always get remainder one.

So it's only divisible so is it a new prime number? Well, there you go, so this is either a new prime number or There may be a number that divides it.

It's got to be a prime which divides it.

But it's not one of those.

So actually you missed some.

So there is another prime out there.

Yeah, you missed one.

Brilliant.

That is an act of genius, Alan.

Is it? Absolutely.

Because that is I must be learning something.

You got to the heart of the problem.

It doesn't matter how many primes you multiply together.

By adding one you'll always discover either a new prime number or a number divisible by a prime that's not on your list.

So with this beautiful argument, Euclid's shown why the primes go on forever but he hasn't helped you to find them.

But I think what's so amazing about this, Euclid proved this 2,000 years ago, and it's as true today when we're sitting on this beach here as it was when he proved it on the beach in Alexandria.

There's something about the permanence of proof that gives you a 100% certainty, and that proof will be there forever.

I mean, this beach will disappear, crumble into sand, but that proof is indestructible.

It sort of gives you a little bit of immortality.

Euclid's name will somehow live forever because of this proof.

It's the beauty of mathematics, this kind of power of proof.

Well, clever Euclid.

He was a clever guy.

I feel pretty confident with prime numbers now.

Yeah.

It does make you want to know more, that's for sure.

The more he talks to you, the more you want to go, "And then what?" It's very enthusiastic, it's passionate, it's amazing.

I don't know where that comes from in a man.

But I much prefer it to the downtrodden apathy of the rest of you.

Prime numbers don't just allow us a glimpse of the infinite.

Recently we've discovered a strange new connection between primes and fundamental properties of the natural world.

To understand this connection, Alan will need to get to grips with some really challenging mathematics.

He's been coping well so far, but what we're about to do is several degrees harder.

OK.

If it's anything to do with primes, I'm ready.

In the 19th century, mathematicians looked at the apparently random distribution of the prime numbers they'd discovered and realised there was some order to them.

A German mathematician called Bernhard Riemann then came up with an exceedingly complicated equation which allowed him to predict the distribution of the primes using this strange three-dimensional graph.

The contours of Riemann's graph appear to expose the hidden pattern of the primes that eluded mathematicians for centuries.

But a recent breakthrough in analysing this graph has unearthed some startling parallels between the abstract world of the primes and the physical world that surrounds us.

Which is why Marcus has brought me here to the National Physical Laboratory.

We're going to undertake an experiment.

Well, Alan, what we are going to do here is we're going to strike this quartz sphere with a ball-bearing ball.

That doesn't sound very technical.

It's not very technical.

I like this kind of physics.

But it is effective! Underneath there are contact microphones, and they convert the vibrations into an electrical signal.

OK, good.

This sounds like quite a straightforward experiment.

I guess it is, but I guess the consequences are more important than that.

This thing is vibrating It'd better be, otherwise you're wasting this man's time.

I feel a lot of pressure.

This is more than darts! I've got to hit that flat bit.

Try your best, Alan.

I've got to get I'll get quite close to it, to make sure.

Good, wow! Now look at that.

Look what I did! Look what you've done.

I threw it quite hard there.

You did! It almost went off the scale.

Sorry.

But every little peak here corresponds to a vibration of this quartz sphere, and the crucial thing you notice is that the way in which those are distributed is related to the series of prime numbers.

This means that if you analyse the distribution of the quartz's vibrations and do the same for the contours of Riemann's graph, you'll see a remarkably similar result.

So similar it cannot be a coincidence.

So some naturally occurring thing in nature that links us PHONE RINGS Is that Riemann? If only I'd known about the quartz! Is there a dial a friend? I'd like to dial a friend.

I did all those sums for years, and I just had to throw a ball-bearing at a bit of quartz! But it's in nature, it's naturally occurring and yet it coincides with the work of this genius mathematician from 200 years ago.

And it's similar enough for you to think, are we on to something? Exactly, similar enough that this can't be a coincidence.

A number pattern's so similar to naturally occurring phenomena, if you like, that maybe we are onto something.

Yeah.

I think this is this is not just a coincidence.

It's a message telling us that the maths which is going on behind this, the way this quartz is vibrating, will help us to explain the way the primes are distributed.

That odd little list of weird looking numbers that don't appear to relate to one another, it's not very pleasant to look at because we like things to go "two, four, six, eight" is actually This is where the pattern is.

They're found everywhere, and they could be the key to everything.

THEY LAUGH This mysterious connection is found elsewhere too.

On a microscopic scale, quantum energy levels in atoms like uranium match this pattern.

Bizarrely, a parallel also exists at the municipal level, in the distribution of bus arrival times in a little-known Mexican city.

There's even a connection between Riemann's graph and the distribution of parked cars in modern-day London.

I think I've converted, I've joined the maths club.

I'm ready now to go forth and do battle with the greatest maths problems of the 21st century.

If only I knew what they were! So, Marcus, what are the big maths questions that we can go after? You want to go for really big iconic problems? Yes! You can't do better than one of the seven millennium problems.

These are the really big questions of maths.

You can win a million dollars for solving any one of these problems.

That's how important they are.

A million dollars for solving a maths equation? Well, slightly more than just a maths equation, but a million dollars.

How does that sound? I'm up for it.

I can see, you're incentivised now! Cos a million divided by two is half a million.

Half a million.

I can see your maths brain is working.

Who said we were going 50/50? OK, ladies and gentlemen, step right up, just a few simple questions.

The Riemann Hypothesis, Navier-Stokes Equations, P v NP.

Try your hand at the Yang-Mills theory.

The Birch and Swinnerton-Dyer, the Hodge, or the Poincare Conjecture.

Don't be shy.

Give it a try! So Marcus, has anyone ever won the million dollars? Well, there's a guy who might be on the verge of winning one.

Grigory Perelman might have solved one of the problems that people have been thinking about for millennia, namely, what is the possible shape of the universe? But if he gets offered the million, it looks like he might actually turn it down.

He's been offered some prizes for solving this problem, and he's turned those down.

So why is he going to turn it down? Well, he's more interested in the buzz of solving maths problems than winning a million dollars.

You can spend a million dollars very quickly but that His name will live forever for solving this iconic problem called the Poincare Conjecture.

That's good for him.

As an amateur mathematician I'm in it for the prizes.

Here you are.

Oh, dear, let's see whether I can That's five with one! He's done it again.

I am a bit of a maths novice, admittedly.

I'm going to see if I can add my pennies-worth to this million-dollar shape of the universe question.

This has to be the biggest problem I've yet tackled.

Trying to imagine the shape of the universe is a problem that mathematicians have been battling with for thousands of years.

So what's the shape? I don't really see a shape there.

I just feel it just goes on and on and on and on and on and on and on.

Well, that's a model, you know.

It's infinitely big.

Stretches out to infinity, I think that's one most people think about.

Does it have an edge, a hard edge? One of the ancient ones was that it was contained in a shell and you'd hit some sort of wall.

Shaped a bit like Russell Grant? Well, maybe.

I bet it is.

But mathematicians have come up with a third idea, which I think is much more satisfying to explain what the shape of the universe is.

To explain this to Alan, I was going to take him back in time.

Well, at least to the 1970s.

If this counts as maths, I should have a PhD.

I spent my childhood playing this Asteroids game.

So this little universe that our spaceman is flying around in will be really useful for trying to understand what the shape of our universe is.

You could say it's just a rectangle.

But it's more interesting than that, because the top of the screen, when your spaceship shoots off the top, it doesn't bounce back again.

It comes back in the bottom.

So actually the bottom of the screen is joined to the top of the screen.

So what shape am I making by joining the top and the bottom of the screen up? A kind of cylinder.

Yeah, exactly.

The screen is wrapped up round into a cylinder.

So the two-dimensional universe gets wrapped round in three dimensions into a cylinder, but not only that What about when I go off stage left, where do you come? You come in stage right.

OK.

So actually the two sides, two ends of the cylinder are also joined.

So if I join these two ends of the cylinder, what shape do I get? OK, now And there's a hint in your brown paper bag.

What I brung from the pier.

Exactly.

It may be relevant.

It is.

The shape of this universe is actually a doughnut.

OK.

So the Asteroids spaceship flies around a two-dimensional universe wrapped into a three-dimensional doughnut shape.

But we live in a three-dimensional universe.

We can move left, right, up, down, and in and out.

So, you know, what's the shape of our universe? Well, this cinema could be a model of our universe.

It's a finite universe, but it could be a universe without any boundaries.

So maybe when I walk this way, and then I hit this wall here, instead of hitting it, maybe I go through and reappear on this side of the universe.

If I go through the screen, maybe I don't hit the screen but I reappear at the back of the cinema.

I've joined up the sides of the cinema, the back and the front and I've got a third dimension.

I could shoot out the top and reappear through the floor.

So this finite universe could have no boundaries just like the Atari game.

But then what's the shape of our little universe that we've created in the cinema? I could start trying to join this cinema up.

We could use this wall here.

We've got to join with the opposite wall, so we could wrap that round, to start making our large doughnut shape, but then I'm going to get stuck.

How do I join the screen with the back of the cinema, and then there's the ceiling with the floor? You're going to need another dimension.

Absolutely.

A bigger dimension.

A fourth dimension.

Just like we learnt In Paris.

In Paris.

So if you give me a fourth dimension, then I've got flexibility to join this shape up, and our little mini-cinema universe would become a doughnut in a four-dimensional world.

Are we living in a four-dimensional doughnut? We're living on the surface of that four-dimensional doughnut.

Can you bake such a thing? This is good.

For the first time in my life, I've been thinking real mathematical thoughts, not just boring old sums.

Everything I've learned so far is being brought to bear on this question of the universe.

I've put aside common sense in favour of logic.

I'm thinking about the higher dimensions that contain our own, the nature of infinity, and the importance of proof.

And I've seen the presence of maths in places I never had before.

All in all, it's got me pondering my own questions on the shape of space.

If this new model of the universe means that we could go into space, keep going and eventually end up right back where we started, does it have another even weirder implication? So does the same thing apply to light? Because if light leaves the Earth and goes out all the way across the universe, eventually it's going to come back and hit us in the back of the head.

Absolutely.

Which means that if we look up, we should be able to see ourselves.

Absolutely.

In fact one of those stars out there might be a version, a young version of our solar system, and astronomers are looking now to see whether one of those stars might in fact be our sun.

You've sold me, Marcus.

Yeah? Yeah, you've changed my view of the universe.

That's fantastic.

So is this it now? Am I at the end of my journey? Alan, this is just the beginning.

Are you angling for a series? I've got it! I can see Milton Keynes!