0:00:05 > 0:00:07BOY: 'One for sorrow
0:00:07 > 0:00:10'Two for mirth
0:00:12 > 0:00:14GIRL: 'Three for a wedding
0:00:14 > 0:00:15'And four for death
0:00:15 > 0:00:17BOY: 'Nine for hell.'
0:00:20 > 0:00:22GIRL: '666.'
0:00:25 > 0:00:30Hidden within this cathedral are clues to a mystery,
0:00:30 > 0:00:33something that could help answer
0:00:33 > 0:00:36one of humanity's most enduring questions...
0:00:37 > 0:00:41..why is the world the way it is?
0:00:42 > 0:00:46The 13th-century masons who constructed this place
0:00:46 > 0:00:48had glimpsed a deep truth
0:00:48 > 0:00:51and they built a message into its very walls
0:00:51 > 0:00:55in the precise proportions of this magnificent cathedral.
0:01:00 > 0:01:02To the medieval clergy,
0:01:02 > 0:01:05these divine numbers were created by God.
0:01:08 > 0:01:11But to me, they're evidence of something else,
0:01:11 > 0:01:15a hidden code that underpins the world around us,
0:01:15 > 0:01:20a code that has the power to unlock the laws that govern the universe.
0:01:47 > 0:01:51As a mathematician, I'm fascinated by the numbers
0:01:51 > 0:01:53and patterns we see all around us...
0:02:03 > 0:02:06..numbers and patterns that connect everything
0:02:06 > 0:02:09from fish to circles
0:02:09 > 0:02:12and from our ancient past
0:02:12 > 0:02:14to the far future.
0:02:19 > 0:02:20INDISTINCT COMMENT
0:02:27 > 0:02:29Together they make up the Code...
0:02:32 > 0:02:35..an abstract world of numbers...
0:02:36 > 0:02:43..that has given us the most detailed description of our world we've ever had.
0:02:55 > 0:03:00For centuries, people have seen significant numbers everywhere...
0:03:01 > 0:03:07..an obsession that's left its mark in the stones of this medieval cathedral.
0:03:18 > 0:03:22In the 12th century, religious scholars here in Chartres
0:03:22 > 0:03:26became convinced these numbers were intrinsically linked to the divine...
0:03:31 > 0:03:34..an idea that dates back to the dawn of Christianity.
0:03:36 > 0:03:40The fourth-century Algerian cleric St Augustine believed
0:03:40 > 0:03:44that seven was so special that it represented the entire universe.
0:03:44 > 0:03:48He described how seven embraced all created things
0:03:48 > 0:03:51and ten was beyond even the universe
0:03:51 > 0:03:54because it was seven plus the three aspects of the Holy Trinity -
0:03:54 > 0:03:57Father, Son and Holy Ghost.
0:04:02 > 0:04:1012 was also hugely important, not simply because there are 12 tribes of Israel or 12 disciples of Jesus,
0:04:10 > 0:04:18but because 12 is divisible by one, two, three, four, six and 12 itself,
0:04:18 > 0:04:20more than any other number around it.
0:04:20 > 0:04:23For St Augustine, numbers had to come from God
0:04:23 > 0:04:27because they obey laws that no man can change.
0:04:30 > 0:04:34Around 800 years after St Augustine,
0:04:34 > 0:04:39the 12th-century Chartres School also recognised their significance.
0:04:43 > 0:04:46It's thought that, under their influence, sacred numbers
0:04:46 > 0:04:51were built into the structure of this majestic building.
0:04:54 > 0:04:59Numbers, they believed, held the key to the mystery of creation.
0:05:06 > 0:05:09I've spent my entire working life studying numbers,
0:05:09 > 0:05:13and for me they're more than just abstract entities.
0:05:13 > 0:05:15They describe the world around us.
0:05:15 > 0:05:18Although I don't share their religious beliefs, I can't help
0:05:18 > 0:05:22feeling something in common with the people who built this place.
0:05:22 > 0:05:25I share their awe and wonder at the beauty of numbers.
0:05:25 > 0:05:31For them, those numbers brought them closer to God, but I think they're important for another reason,
0:05:31 > 0:05:35because I believe they're the key to making sense of our world.
0:05:40 > 0:05:46Numbers have given us an unparalleled ability to understand our universe.
0:05:49 > 0:05:54And in places, this code literally emerges from the ground.
0:06:00 > 0:06:03Rural Alabama,
0:06:03 > 0:06:06spring 2011.
0:06:08 > 0:06:11Warm, lush and peaceful.
0:06:17 > 0:06:20But this year, there's a plague coming.
0:06:28 > 0:06:30While some locals are moving out,
0:06:30 > 0:06:35Dr John Cooley has driven thousands of miles to be here.
0:06:39 > 0:06:43He's on the trail of one of the area's strangest residents.
0:06:50 > 0:06:55We have been driving around looking for the emergences for about three and a half weeks.
0:06:55 > 0:07:00I've driven 7,200 miles since Good Friday trying to figure out where these things are.
0:07:05 > 0:07:10What makes these insects so remarkable is their bizarre lifecycle.
0:07:12 > 0:07:18For 12 whole years, they live hidden underground, in vast numbers.
0:07:22 > 0:07:25Then, in their 13th year...
0:07:25 > 0:07:27at precisely the same time...
0:07:29 > 0:07:33..they all burrow out from the earth to breed.
0:07:39 > 0:07:45At the full part of the emergence, there will be millions of insects out per acre. They'll be everywhere.
0:07:45 > 0:07:47It really is insect mayhem.
0:07:54 > 0:07:58This is the periodical cicada.
0:08:00 > 0:08:02This one is a male...
0:08:04 > 0:08:07..and you know that because on the abdomen,
0:08:07 > 0:08:09there's a pair of organs called timbles,
0:08:09 > 0:08:11and they're sound-producing organs.
0:08:11 > 0:08:14It's a little membrane that's vibrated, it makes a sound.
0:08:14 > 0:08:17Oh, yeah. I don't have to be frightened of these, do I?
0:08:17 > 0:08:21- No, no, they're absolutely harmless. They make wonderful pets.- Really?
0:08:21 > 0:08:24- Mm-hm.- They're quite ticklish. - It's a harmless insect.
0:08:24 > 0:08:27It doesn't bite, it doesn't sting, nothing of that sort.
0:08:27 > 0:08:30Its only defence is safety in numbers.
0:08:32 > 0:08:37By emerging in such vast numbers, each individual cicada
0:08:37 > 0:08:39minimises its risk of being eaten.
0:08:39 > 0:08:42Because there are so many of them,
0:08:42 > 0:08:46their predators simply can't eat them fast enough.
0:08:47 > 0:08:50Well, you can certainly hear the cicadas.
0:08:50 > 0:08:53Yes, you can. There are probably millions of them up there.
0:08:53 > 0:08:59- Millions?- Yeah, millions. What you probably don't realise is you're only hearing half the population.
0:08:59 > 0:09:00Only the males make these loud sounds.
0:09:00 > 0:09:03There are just as many females up there as well.
0:09:03 > 0:09:07And it's extraordinary to think that if we came here next year,
0:09:07 > 0:09:10- we wouldn't hear this sound at all? - You'll have to come back in 13 years.
0:09:10 > 0:09:15So 2024 is when you'll hear the forest singing like this again?
0:09:15 > 0:09:17- That's right.- That's amazing.
0:09:24 > 0:09:29Why have the cicadas evolved with this 13-year lifecycle as opposed to any other number?
0:09:29 > 0:09:35Well, you have to remember that these cicadas require large numbers to survive predators,
0:09:35 > 0:09:40and so we think that these long lifecycles in some way help them maintain large populations.
0:09:45 > 0:09:49John believes that, by appearing every 13 years,
0:09:49 > 0:09:52the cicadas minimise their chances of emerging at the same time
0:09:52 > 0:09:56as other cicadas with different lifecycles...
0:09:58 > 0:10:04..because if they were to interbreed, it could have disastrous consequences.
0:10:06 > 0:10:10The offspring would have unusual lifecycles.
0:10:10 > 0:10:15They're going to emerge a little bit here, a little bit there, some this year and some that year in small
0:10:15 > 0:10:20numbers, and that's key because if they emerge in small numbers, the predators eat them.
0:10:32 > 0:10:37The cicadas' survival depends on avoiding other broods.
0:10:52 > 0:10:57Imagine you've got a brood of cicadas that appears every six years.
0:11:09 > 0:11:12Now, let's suppose there's another brood
0:11:12 > 0:11:15which wants to try and avoid the red cicadas.
0:11:15 > 0:11:20One way to do that would be to appear less often in the forest, and that actually works.
0:11:20 > 0:11:24So let's suppose this brood appears every nine years.
0:11:32 > 0:11:35So if the green cicada appears every nine years,
0:11:35 > 0:11:39then it only coincides with the red cicada every 18 years.
0:11:40 > 0:11:45But, rather surprisingly, a smaller number, seven, works even better.
0:11:56 > 0:12:00Coming out every seven years instead of every nine
0:12:00 > 0:12:03means the cicadas appear together much less often.
0:12:06 > 0:12:10Now they only coincide every 42 years.
0:12:11 > 0:12:14That's just twice every century.
0:12:17 > 0:12:19And for the real cicadas,
0:12:19 > 0:12:25a 13-year lifecycle has exactly the same effect as seven does here
0:12:25 > 0:12:30because they both belong to a special series of numbers.
0:12:30 > 0:12:34Like 13, seven is a prime number.
0:12:34 > 0:12:39Unlike other numbers, primes can only be divided by themselves and one,
0:12:39 > 0:12:44and it's this property that means that numbers that are separated by primes
0:12:44 > 0:12:47are far less likely to coincide with multiples of other numbers.
0:12:49 > 0:12:54Because 13 is a prime number, a 13-year lifecycle
0:12:54 > 0:12:58makes the cicadas much less likely to coincide with other groups.
0:13:01 > 0:13:05Up in Georgia, there is another brood of periodical cicada
0:13:05 > 0:13:07and they, too, have a prime number lifecycle.
0:13:07 > 0:13:10They come out every 17 years.
0:13:10 > 0:13:15Because 13 and 17 are both prime numbers,
0:13:15 > 0:13:21the two broods only emerge together once every 221 years.
0:13:28 > 0:13:33Prime numbers are intimately linked to the cicadas' survival
0:13:33 > 0:13:35and, intriguingly,
0:13:35 > 0:13:40they're one of the most important elements of the Code,
0:13:40 > 0:13:44because the Code is a mathematical world,
0:13:44 > 0:13:48built from numbers.
0:13:48 > 0:13:52Just as atoms are the indivisible units that make up every physical object,
0:13:52 > 0:13:57so prime numbers are the indivisible building blocks of the Code.
0:14:02 > 0:14:06Prime numbers are indivisible, which means they can't be made
0:14:06 > 0:14:09by multiplying any other numbers together.
0:14:11 > 0:14:16But every non-prime number can be created by multiplying primes together.
0:14:19 > 0:14:22It's impossible to make any numbers without them.
0:14:28 > 0:14:32And if any primes are missing,
0:14:32 > 0:14:35there will always be some numbers you can't create.
0:14:42 > 0:14:47For me, the fact that the most fundamental units of mathematics
0:14:47 > 0:14:49can be found woven into the natural world
0:14:49 > 0:14:54is not only compelling evidence that the Code exists,
0:14:54 > 0:14:58but also that numbers underpin everything...
0:14:59 > 0:15:03..including our own biology.
0:15:33 > 0:15:36This is an innately human characteristic.
0:15:36 > 0:15:42Music is one of the things which defines who we are, and each culture has its own particular style.
0:15:42 > 0:15:45These guys make it seem so effortless, as if the notes
0:15:45 > 0:15:49are just thrown together, but that's simply an illusion.
0:15:52 > 0:15:56MUSIC ENDS, APPLAUSE
0:15:57 > 0:16:01Because, just as numbers govern the cicadas' lives,
0:16:01 > 0:16:04so they determine how WE hear sound.
0:16:25 > 0:16:27That's a C.
0:16:27 > 0:16:30And using this oscilloscope, I can get a picture of that note.
0:16:31 > 0:16:34So I can actually SEE the sound wave.
0:16:34 > 0:16:38Now, the height of the wave corresponds to how loudly I'm playing the note,
0:16:38 > 0:16:41so if I play the note very quietly...
0:16:41 > 0:16:46play it very loudly...I suddenly get a huge wave on the screen.
0:16:46 > 0:16:50The more important thing is the distance between the peaks of the wave,
0:16:50 > 0:16:53because that's determined by the pitch or frequency of the note.
0:16:54 > 0:16:56'The higher the note...
0:16:57 > 0:17:01'the shorter the distance between the peaks.'
0:17:08 > 0:17:10Now, look what happens when I play a C...
0:17:12 > 0:17:16..and compare that with the same note, a C, but an octave higher.
0:17:19 > 0:17:22Something rather surprising emerges,
0:17:22 > 0:17:26because now you can see that the higher note has twice
0:17:26 > 0:17:28as many peaks as the lower note,
0:17:28 > 0:17:34which means the frequency of the high C is twice that of the low C.
0:17:34 > 0:17:37And this happens whatever two notes you choose.
0:17:37 > 0:17:43Provided they're an octave apart, then their frequencies are going to be in this one-to-two ratio.
0:17:47 > 0:17:52Two notes which are an octave apart just sound nice together, and they're actually the most
0:17:52 > 0:17:55harmonious combination of notes that you can have.
0:17:55 > 0:18:01And that's because one to two is the simplest possible frequency relationship, and that's what
0:18:01 > 0:18:08music is all about, because it's these simple whole-number ratios that sound so good to the ear.
0:18:08 > 0:18:10A perfect fifth...
0:18:10 > 0:18:13is a frequency ratio of three to two.
0:18:13 > 0:18:15A perfect fourth...
0:18:15 > 0:18:16is four to three.
0:18:16 > 0:18:20And a slightly more complex sound, a minor sixth...
0:18:22 > 0:18:25..that's a frequency ratio of five to eight.
0:18:27 > 0:18:33Every combination of notes used in music is defined by simple ratios.
0:18:35 > 0:18:39Although we might not be aware of it, these numerical rules underpin
0:18:39 > 0:18:45everything from the simplest song to the most elaborate symphony.
0:18:45 > 0:18:49They're so deeply ingrained that when they're broken,
0:18:49 > 0:18:52we intuitively know something is wrong.
0:19:04 > 0:19:08Professor Judy Edworthy understands this more than most.
0:19:12 > 0:19:19She spends her time subjecting people to some of most unpleasant noises imaginable.
0:19:20 > 0:19:21Hi, Judy.
0:19:21 > 0:19:23- Ah, hello.- Marcus.
0:19:23 > 0:19:27'Her research investigates the psychological effects of sound.
0:19:32 > 0:19:40'And by using complex ratios instead of simple ones, the noises she creates are nothing like music.'
0:19:40 > 0:19:44You can see just by looking at it it's not going to sound nice.
0:19:44 > 0:19:46The wave looks a mess.
0:19:46 > 0:19:48The wave is a mess. It's very difficult to see a pattern.
0:19:48 > 0:19:53CONSTANT DRONE
0:19:53 > 0:19:55OK. It sounds really quite odd now.
0:19:55 > 0:20:00It doesn't have any pitch. It sounds harsh and I could make it louder and that would make it harsher.
0:20:00 > 0:20:04When the various frequencies aren't simple multiples of one another,
0:20:04 > 0:20:06there's no common pattern for the ear to respond to,
0:20:06 > 0:20:11and the more complex you make the ratios, the more dissonant and harsh the sound will get.
0:20:14 > 0:20:19By monitoring her victims' reactions to these appalling noises,
0:20:19 > 0:20:22Professor Edworthy has found they have a very different effect
0:20:22 > 0:20:24on our minds than music.
0:20:24 > 0:20:26ALARM BEEPS
0:20:26 > 0:20:29HONKING
0:20:29 > 0:20:30WHIRRING
0:20:30 > 0:20:32- They're so unpleasant... - HAMMERING
0:20:32 > 0:20:36..they shock our brains into action.
0:20:36 > 0:20:37For example, a siren.
0:20:37 > 0:20:41HIGH-PITCHED SIREN BLARES
0:20:44 > 0:20:49That's quite a harsh sound, but it's designed for a purpose - to get you out of the way.
0:20:49 > 0:20:52Sometimes you find these sounds in the animal world as well.
0:20:52 > 0:20:55So this, for example, this is a chimpanzee and an orang-utan.
0:20:55 > 0:20:59INTERMITTENT SCREECHING
0:21:02 > 0:21:06OK, these animals are obviously quite bothered by something.
0:21:06 > 0:21:10You don't need to know what that sound means to know that that animal's not happy
0:21:10 > 0:21:16and also that the other animals in that environment and us, for example, should just get out of the way.
0:21:16 > 0:21:18SHORT SCREECH
0:21:18 > 0:21:22So it's interesting that we really hear pattern,
0:21:22 > 0:21:26and when it isn't there, it creates an effect in all of us.
0:21:26 > 0:21:29LOW-PITCHED SCREECH
0:21:36 > 0:21:40Remarkably, it's numerical patterns in the Code
0:21:40 > 0:21:44that dictate the combinations of sounds we hear as music...
0:21:44 > 0:21:46RUSTLING
0:21:46 > 0:21:49- ..and those we hear simply as noise. - CHIRPING, SIREN
0:21:49 > 0:21:53BELL TOLLS
0:21:53 > 0:21:57And perhaps stranger still, it's these same numbers
0:21:57 > 0:22:01that are built into the walls of this medieval cathedral.
0:22:07 > 0:22:11Two notes which are an octave apart are going be in this one-to-two ratio.
0:22:19 > 0:22:22The width of the nave here is twice the distance between
0:22:22 > 0:22:29each of the columns that run up its length - a ratio of two to one.
0:22:29 > 0:22:33The most harmonious combination of notes from a pair.
0:22:33 > 0:22:37The altar divides the nave into a ratio of eight to five.
0:22:39 > 0:22:41A minor sixth...
0:22:41 > 0:22:43eight to five.
0:22:46 > 0:22:48A perfect fifth...
0:22:48 > 0:22:50three to two.
0:22:50 > 0:22:54A perfect fourth is four to three.
0:22:54 > 0:22:56Major third, five to four.
0:22:59 > 0:23:03And that's what music is all about.
0:23:03 > 0:23:07St Augustine believed these ratios were used by God to construct the universe
0:23:07 > 0:23:12and that that was why they produced harmony in music.
0:23:17 > 0:23:21By constructing their cathedral using the same ratios,
0:23:21 > 0:23:25the clergy at Chartres hoped to echo God's creation.
0:23:25 > 0:23:28This entire place is a symphony set in stone.
0:23:32 > 0:23:37Using the Code's numbers has created a building of awe-inspiring beauty.
0:23:51 > 0:23:53The only truth there is...
0:23:53 > 0:23:56Seemingly significant numbers...
0:24:02 > 0:24:05By searching for divine meaning in numbers,
0:24:05 > 0:24:1012th-century scholars had stumbled across elements of the Code.
0:24:10 > 0:24:13It's very difficult to see a pattern.
0:24:15 > 0:24:20Mysterious numbers and patterns that seem to be written into our biology.
0:24:21 > 0:24:24Its only defence is safety in numbers.
0:24:26 > 0:24:33And as we've looked closer, we haven't simply found more numbers -
0:24:33 > 0:24:40we've begun to uncover their strangest properties and started to see deep connections between them.
0:24:44 > 0:24:48Back in the distant past, in Neolithic times,
0:24:48 > 0:24:54around 4,000 years ago, an ancient people brought these stones here
0:24:54 > 0:24:56and arranged them like this.
0:24:56 > 0:25:01This is Sunkenkirk stone circle in Cumbria and it's one of around 1,000
0:25:01 > 0:25:06such structures that our ancient ancestors built across the UK.
0:25:12 > 0:25:16Stretching back into the mists of time,
0:25:16 > 0:25:19the circle has been steeped in mysticism.
0:25:23 > 0:25:26But whether the people who built this structure knew it or not,
0:25:26 > 0:25:30there is deep significance hidden inside this circle.
0:25:30 > 0:25:34OK, so I need to start by measuring the diameter
0:25:34 > 0:25:40of my circle, so that's the distance from one edge to the other.
0:25:42 > 0:25:45I need to go roughly through the centre.
0:25:47 > 0:25:50So that's 27 and 90.
0:25:54 > 0:25:57Right, so now I'm going to measure the circumference
0:25:57 > 0:26:00of the circle. So off we go.
0:26:00 > 0:26:02So around the outside.
0:26:04 > 0:26:07Oh, I've never got so much exercise doing maths before!
0:26:09 > 0:26:12And that's the circumference.
0:26:12 > 0:26:16So I've got 91 metres
0:26:16 > 0:26:19and 70 centimetres.
0:26:21 > 0:26:27I'm going to do a little calculation. I'm going to divide the circumference
0:26:27 > 0:26:31of the circle by the diameter.
0:26:31 > 0:26:35So 917 divided by 279.
0:26:35 > 0:26:37So that's roughly three...
0:26:37 > 0:26:41Bit of, er, mental arithmetic, not a mathematician's strongest point.
0:26:41 > 0:26:43OK, two lots of 279,
0:26:43 > 0:26:45so...
0:26:45 > 0:26:48not far out from what I was hoping for.
0:26:48 > 0:26:53So when I do that, I get roughly 3.2 as the answer.
0:26:58 > 0:27:02My measurements weren't very precise...
0:27:03 > 0:27:09..but my answer is close to a mysterious number hidden within every circle.
0:27:13 > 0:27:18So, for example, let's take this circular plate here.
0:27:18 > 0:27:20I'm going to measure its diameter.
0:27:20 > 0:27:2426.4 centimetres. Now its circumference.
0:27:26 > 0:27:28That's a bit trickier.
0:27:28 > 0:27:3082.9 centimetres.
0:27:30 > 0:27:34Divide the circumference by the diameter, I get 3.14.
0:27:34 > 0:27:37Now let's take another circle. Measure its diameter.
0:27:37 > 0:27:3912.8 centimetres.
0:27:41 > 0:27:46So the circumference is 40.2 centimetres.
0:27:46 > 0:27:51Divide the circumference by the diameter and I get 3.14.
0:27:51 > 0:27:55In fact, whatever circle I take, divide the circumference
0:27:55 > 0:28:00by the diameter and you're going to get a number which starts 3.14.
0:28:00 > 0:28:03This is a number we call pi.
0:28:08 > 0:28:12No matter where the circles are, no matter how big or small...
0:28:14 > 0:28:16..they will always contain pi.
0:28:19 > 0:28:25It's this universality of the number pi which tells you you've identified a piece of true Code.
0:28:25 > 0:28:27In fact, if you get another number,
0:28:27 > 0:28:29it means that you haven't got a circle.
0:28:29 > 0:28:32In some sense, pi is the essence of circleness,
0:28:32 > 0:28:35distilled into the language of the Code.
0:28:36 > 0:28:42And because circles and curves crop up again and again in nature,
0:28:42 > 0:28:46pi can be found all around us.
0:28:50 > 0:28:53It's in the gentle curve of a river...
0:28:54 > 0:28:57..the sweep of a coast line...
0:28:59 > 0:29:03..and the shifting patterns of the desert sands.
0:29:05 > 0:29:11Pi seems written into the structures and processes of our planet.
0:29:17 > 0:29:21But, strangely, pi also appears in places
0:29:21 > 0:29:24that seem to have nothing to do with circles.
0:29:30 > 0:29:34I started fishing Brighton in 1972.
0:29:34 > 0:29:38I've been a fisherman 40 years, catching Dover sole.
0:29:40 > 0:29:44That's the main target species for the English Channel.
0:29:45 > 0:29:47How many fish do you think you get a day?
0:29:47 > 0:29:49300 some days, 150 other days,
0:29:49 > 0:29:51so I'd say 200 would be average.
0:29:51 > 0:29:56And you've got me some Dover sole today so I can have a weigh of what you've caught today.
0:29:56 > 0:29:58- Yeah, you can play with them!- OK!
0:30:01 > 0:30:05What's remarkable is that, with just a small amount of information...
0:30:05 > 0:30:07It's 180 grams.
0:30:08 > 0:30:10..and by weighing a few fish...
0:30:10 > 0:30:12That's a whopper.
0:30:12 > 0:30:13..I can use the Code
0:30:13 > 0:30:16to tell me things about not just today's catch...
0:30:16 > 0:30:20360 grams. 50 grams. 110 grams.
0:30:20 > 0:30:24..but about all the Dover sole Sam's ever fished...
0:30:24 > 0:30:26Whoa, jeez, come back!
0:30:26 > 0:30:29..I can even get an estimate for the largest sole
0:30:29 > 0:30:32that Sam is likely to have caught during his career.
0:30:32 > 0:30:33Right...
0:30:33 > 0:30:37First , I need to work out what the average weight of a fish is,
0:30:37 > 0:30:40or the mean,
0:30:40 > 0:30:44so 140 plus 190
0:30:44 > 0:30:46plus 150...
0:30:46 > 0:30:51So now I need to work out the standard deviation, so that's 140 minus square that...
0:30:51 > 0:30:54Bear with me, all right? Almost there.
0:30:54 > 0:30:59So he said he fished for 40 years,
0:30:59 > 0:31:04and eight weeks during the year, six days out of the week
0:31:04 > 0:31:09and 200 sole each day,
0:31:09 > 0:31:13so that gives you a total of 384,000 fish.
0:31:14 > 0:31:18Using these numbers, I can calculate that the largest one
0:31:18 > 0:31:21out of those 384,000 fish
0:31:21 > 0:31:26should be about 1.3 kilograms, which is roughly three pounds.
0:31:28 > 0:31:33So what's the largest Dover sole that you've caught in your career?
0:31:33 > 0:31:36We call them door mats, the large ones,
0:31:36 > 0:31:39and you maybe get four or five a season.
0:31:39 > 0:31:44The largest, I'd say, was three to three and a half pounds.
0:31:44 > 0:31:48An average Dover Sole is that sort of size
0:31:48 > 0:31:50and these...
0:31:50 > 0:31:53Wow, that's huge! Yeah!
0:31:53 > 0:31:57It's a whopper. It's always nice to catch big stuff, you know.
0:31:57 > 0:31:59Well, I think it is anyway. HE CHUCKLES
0:32:04 > 0:32:07Using the Code, it's possible to estimate the size
0:32:07 > 0:32:10of the biggest fish Sam's ever caught,
0:32:10 > 0:32:15despite not weighing a single fish anywhere near that size.
0:32:19 > 0:32:26Now, the reason this calculation is possible is because the distribution of the weights of fish,
0:32:26 > 0:32:31in fact the distribution of lots of things like the height of people in the UK or IQ,
0:32:31 > 0:32:34is given by this formula.
0:32:34 > 0:32:37'This is the normal distribution equation,
0:32:37 > 0:32:41'one of the most important bits of mathematics
0:32:41 > 0:32:45'for understanding variation in the natural world.'
0:32:45 > 0:32:50The most remarkable thing about this formula isn't so much what it does
0:32:50 > 0:32:52as this term here, pi.
0:32:52 > 0:32:54It seems totally bizarre
0:32:54 > 0:32:58that a bit of the Code that has something to do with the geometry of a circle
0:32:58 > 0:33:01can help you to calculate the weight of fish.
0:33:01 > 0:33:06Pi shouldn't have anything to do with fish, yet there it is.
0:33:14 > 0:33:18Just as the circle appears everywhere in nature,
0:33:18 > 0:33:23so pi crops up again and again in the mathematical world.
0:33:24 > 0:33:30It's an astonishing example of the interconnectedness of the Code.
0:33:30 > 0:33:35A glimpse into a world where numbers don't just have strange connections,
0:33:35 > 0:33:39they have deeply puzzling properties of their own.
0:33:42 > 0:33:45Pi is what's known as an irrational number.
0:33:47 > 0:33:51Written as a decimal, it has an infinite number of digits
0:33:51 > 0:33:55arranged in a sequence that never repeats.
0:33:57 > 0:34:01And it's thought that any number you can possibly imagine
0:34:01 > 0:34:06will appear in pi somewhere, from my birthday
0:34:06 > 0:34:10to the answer to life, the universe and everything.
0:34:13 > 0:34:16Because they go on for ever, we can never know all the digits
0:34:16 > 0:34:17that make up pi.
0:34:17 > 0:34:21But, luckily, we only need the first 39
0:34:21 > 0:34:26to calculate the circumference of a circle the size of the entire observable universe,
0:34:26 > 0:34:30accurate to the radius of a single hydrogen atom.
0:34:37 > 0:34:42But as strange as Pi is, it does at least describe a physical object.
0:34:43 > 0:34:47Some numbers don't make any sense in real world,
0:34:47 > 0:34:50despite the fact we use them all the time.
0:34:50 > 0:34:53Numbers, like negative numbers.
0:34:55 > 0:35:00It's impossible to trade anything, stocks, shares, currency,
0:35:00 > 0:35:03even fish, without negative numbers.
0:35:03 > 0:35:05Most of us are comfortable them.
0:35:05 > 0:35:08Even though we may not like it, we understand what it means
0:35:08 > 0:35:10to have a negative bank balance.
0:35:10 > 0:35:12But when you start to think about it,
0:35:12 > 0:35:16there's something deeply strange about negative numbers,
0:35:16 > 0:35:20cos they don't seem to correspond to anything real at all.
0:35:22 > 0:35:27The deeper we look into the Code, the more bizarre it becomes.
0:35:32 > 0:35:39It's easy to imagine one fish or two fish, or no fish at all.
0:35:39 > 0:35:43It's much harder to imagine what minus-one fish looks like.
0:35:43 > 0:35:48Negative numbers are so odd that if I have minus-one fish and you give me a fish,
0:35:48 > 0:35:52then all you can be certain of is that I've got no fish at all.
0:35:59 > 0:36:05Numbers, can exist regardless of whether they make any sense in the physical world.
0:36:09 > 0:36:14And if you think that's odd, some numbers are so strange
0:36:14 > 0:36:17they don't even seem to make sense as numbers.
0:36:18 > 0:36:22Now, this is one of the most basic facts of mathematics.
0:36:22 > 0:36:27A positive number multiplied by another positive number is a positive number.
0:36:27 > 0:36:33So for example, one times one is one.
0:36:33 > 0:36:37A negative number multiplied by another negative number
0:36:37 > 0:36:40also gives a positive number.
0:36:40 > 0:36:46So for example, minus-one times minus-one is plus-one.
0:36:46 > 0:36:52'It's not only a rule, it's a proven truth of multiplication.
0:36:52 > 0:36:56'Whenever the signs are the same, the product is always positive.'
0:36:56 > 0:36:58From this, it's obvious
0:36:58 > 0:37:00if I take any number and multiply it by itself,
0:37:00 > 0:37:03then the answer is going to be positive.
0:37:03 > 0:37:05However, in the Code,
0:37:05 > 0:37:08there's a special number which breaks this rule.
0:37:08 > 0:37:12When I multiply it by itself, it gives the answer minus-one.
0:37:12 > 0:37:16It's impossible to imagine what this number could be,
0:37:16 > 0:37:19because there simply is no number
0:37:19 > 0:37:23that when multiplied by itself, gives minus-one.
0:37:23 > 0:37:28This isn't a number I can calculate. I can't show you this number.
0:37:28 > 0:37:30Nevertheless, we've given this number a name.
0:37:30 > 0:37:33It's called "i", and it's part of a whole class of new numbers
0:37:33 > 0:37:35called imaginary numbers.
0:37:37 > 0:37:41Calculating with imaginary numbers is the mathematical equivalent
0:37:41 > 0:37:43of believing in fairies.
0:37:45 > 0:37:49But even these strangest elements of the Code turn out to have
0:37:49 > 0:37:52some very practical applications.
0:37:56 > 0:38:00The ground's close, will you call me, please, 1-1-9 next...
0:38:03 > 0:38:08Runway 25, clear to land. Surface is 1-3-0, less than five minutes.
0:38:08 > 0:38:11'Especially on a day like this.'
0:38:14 > 0:38:198-5 Foxtrot, thank you, vacate next right and park yourself 1-3 short.
0:38:19 > 0:38:24'8-5 Foxtrot, 8-2-0, both making approach down direct and right, 2-5.'
0:38:24 > 0:38:26So where's this one coming from?
0:38:26 > 0:38:30That is from Barcelona. It's an Easyjet flight, EZZ6402.
0:38:30 > 0:38:34Don't know how many people are on board, but it seats about 190.
0:38:34 > 0:38:36And here he is. He's getting pretty close now.
0:38:36 > 0:38:38Just less than two miles till he lands.
0:38:38 > 0:38:42What information is the radar giving you about the aeroplanes?
0:38:42 > 0:38:46The first and most important thing is the position of the aircraft.
0:38:46 > 0:38:49The yellow slash there is where the aircraft is.
0:38:49 > 0:38:53You've got the blue trail, the history of where the aircraft's been.
0:38:53 > 0:38:58From that you get two things - you get its rough heading, where he's going, and its speed.
0:38:58 > 0:39:00The longer the trail, the faster the aircraft's going.
0:39:06 > 0:39:09Radar works by sending out a pulse of radio waves
0:39:09 > 0:39:13and analysing the small fraction of the signal that's reflected back.
0:39:17 > 0:39:21Complex computation is then needed to distinguish moving objects,
0:39:21 > 0:39:25like planes, from the stationary background.
0:39:25 > 0:39:28RADIO COMMUNICATION
0:39:28 > 0:39:35At the heart of that analysis lies "i", the number that cannot exist.
0:39:36 > 0:39:41Imaginary numbers are useful for working out the complex way
0:39:41 > 0:39:43radio waves interact with each other.
0:39:43 > 0:39:47It seems to be the right language to describe their behaviour.
0:39:47 > 0:39:50Now, you could do these calculations with ordinary numbers.
0:39:50 > 0:39:52But they're so cumbersome,
0:39:52 > 0:39:56by the time you've done the calculation the plane's moved to somewhere else.
0:39:56 > 0:40:00Attitude 6,000 on a squawk of 7-7-1-5.
0:40:00 > 0:40:03Using imaginary numbers makes the calculation simpler
0:40:03 > 0:40:06that you can track the planes in real time.
0:40:06 > 0:40:11In fact without them, radar would be next to useless for Air Traffic Control.
0:40:15 > 0:40:19It's kind of amazing that this abstract idea lands planes.
0:40:19 > 0:40:22It's a bit surprising, you're talking about imaginary numbers
0:40:22 > 0:40:24- and this isn't imaginary, this is real.- This is very real.
0:40:24 > 0:40:28I'm surprised at the fact that something so abstract
0:40:28 > 0:40:30is being used in such a concrete way.
0:40:45 > 0:40:48As strange as it may seem, the code provides us
0:40:48 > 0:40:52with an astonishingly successful description of our world.
0:40:58 > 0:41:03Its most ethereal numbers have starkly real applications.
0:41:03 > 0:41:09Its patterns can explain one of the most profound processes in nature -
0:41:09 > 0:41:12how living things grow.
0:41:15 > 0:41:18This is a picture of something I've been fascinated by
0:41:18 > 0:41:20ever since I became a mathematician.
0:41:20 > 0:41:25It's an X-ray of a marine animal called a nautilus.
0:41:25 > 0:41:30And this spiral here is one of the iconic images of mathematics.
0:41:30 > 0:41:33Now, while I've seen pictures like this hundreds of times,
0:41:33 > 0:41:36I've never actually seen the animal for real.
0:41:39 > 0:41:44'At Brooklyn College, biologist Jennifer Basil keeps five of these aquatic denizens,
0:41:44 > 0:41:48'for her research into the evolution of intelligence.'
0:41:50 > 0:41:54We keep the animals in these tall tanks because they're naturally active at night
0:41:54 > 0:41:58and they like darkness, they live in deep water.
0:41:58 > 0:42:00They also like to go up and down in the water column,
0:42:00 > 0:42:02- that kind of makes them happy.- OK!
0:42:02 > 0:42:05- We give them the five-star treatment here.- Right...
0:42:07 > 0:42:10- This is Number Five.- Ah, wow.- Yeah.
0:42:10 > 0:42:11Gosh, big eyes.
0:42:11 > 0:42:15- They have huge eyes, great for seeing in low light conditions.- Right.
0:42:16 > 0:42:18- So, here's that beautiful shell. - Yeah.
0:42:18 > 0:42:22And the striping pattern helps them hide where they live.
0:42:38 > 0:42:43I've never seen the animal before inside the shell, what is it?
0:42:43 > 0:42:46They're related to octopuses, squids and cuttlefish.
0:42:46 > 0:42:49It's a little bit like an octopus with a shell
0:42:49 > 0:42:53and what's amazing about them is that their lineage
0:42:53 > 0:42:57is hundreds of millions of years old and they haven't changed very much
0:42:57 > 0:42:59in all that time. We call them a living fossil.
0:42:59 > 0:43:04It's a great opportunity to look at an ancient brain and behaviour
0:43:04 > 0:43:07and they're a wonderful way to study the evolution of intelligence.
0:43:07 > 0:43:10So are these guys intelligent, then?
0:43:10 > 0:43:14Some are smarter than others, like that's Number Four,
0:43:14 > 0:43:17he outperforms everybody in all the memory tests.
0:43:17 > 0:43:20He's quite active all the time, he's quite engaging.
0:43:20 > 0:43:22If you put your in the water he comes up to you,
0:43:22 > 0:43:26whereas Number Three, who happens to be a teenager,
0:43:26 > 0:43:29is I'd guess you'd say more shy and you put him in a new place
0:43:29 > 0:43:33and he sort of just attaches to the wall and sits there.
0:43:33 > 0:43:36I'm interested in the shell as a mathematician,
0:43:36 > 0:43:38but what does the nautilus use the shell for?
0:43:38 > 0:43:41I think the most obvious use is protection.
0:43:42 > 0:43:44They also use it for buoyancy.
0:43:44 > 0:43:46They only live in the front chamber
0:43:46 > 0:43:49and all the other chambers are filled with gas
0:43:49 > 0:43:50and with some fluid.
0:43:50 > 0:43:55By regulating that, they can gently and passively move up and down
0:43:55 > 0:43:57in the water like a submarine.
0:43:57 > 0:43:59The really cool thing they can do
0:43:59 > 0:44:03is they can actually survive on the oxygen in the chambers,
0:44:03 > 0:44:07if there's a period where the oxygen goes down in the oceans.
0:44:07 > 0:44:11It's one of the reasons why they've lived for millions of years.
0:44:11 > 0:44:14It's a really great adaptation. The shell is really amazing.
0:44:17 > 0:44:21But perhaps even more remarkably, the rules this ancient creature
0:44:21 > 0:44:23uses to construct its home
0:44:23 > 0:44:27are written in the language of the Code.
0:44:27 > 0:44:29HORNS BLARE
0:44:36 > 0:44:41The nautilus shell is one of the most beautiful and intricate structures in nature.
0:44:41 > 0:44:44Here you can see the chambers. This is the one where it lives
0:44:44 > 0:44:47and these are the ones it uses for buoyancy.
0:44:47 > 0:44:50Now, at first sight, this looks like a really complex shape,
0:44:50 > 0:44:53but if I measure the dimensions of these chambers
0:44:53 > 0:44:56a clear pattern begins to emerge.
0:45:09 > 0:45:13Now there doesn't seem to be any connection between these numbers,
0:45:13 > 0:45:16but look what happens when I take each number
0:45:16 > 0:45:19and divide it by the previous measurement.
0:45:19 > 0:45:25If I take 3.32 and divide by 3.07,
0:45:25 > 0:45:27I get 1.08.
0:45:27 > 0:45:30Divide 3.59 by 3.32
0:45:30 > 0:45:33and I get 1.08.
0:45:33 > 0:45:37Take 3.88 and divide by 3.59 and I get, again, 1.08.
0:45:39 > 0:45:44So every time I do this calculation, I get the same number.
0:45:44 > 0:45:46So although it's not clear by looking at the shell,
0:45:46 > 0:45:51this tells us that the nautilus is growing at a constant rate.
0:45:51 > 0:45:55Everytime the nautilus builds a new room, the dimensions of that room
0:45:55 > 0:45:58are 1.08 times the dimensions of the previous one.
0:45:58 > 0:46:02And it's just by following this simple mathematical rule
0:46:02 > 0:46:05that the nautilus builds this elegant spiral.
0:46:08 > 0:46:12And because many living things grow in a similar way,
0:46:12 > 0:46:15these spirals are everywhere.
0:46:17 > 0:46:22The rules nature uses to create its patterns are found in the Code.
0:46:50 > 0:46:55Behind the world we inhabit, there's a strange and wonderful mathematical realm.
0:46:55 > 0:46:59They're actually related to octopus, squids and cuttlefish.
0:46:59 > 0:47:00They're quite ticklish.
0:47:04 > 0:47:10The numbers and connections at its heart describe the processes we see all around us.
0:47:10 > 0:47:11Bear with me, all right?
0:47:16 > 0:47:21But the Code doesn't just contain the rules that govern our planet -
0:47:21 > 0:47:27its numbers also describe the laws that control the entire universe.
0:47:39 > 0:47:44For centuries, we've gazed out into the night's sky
0:47:44 > 0:47:49and tried to make sense of the patterns we see in the stars.
0:48:07 > 0:48:12To take a closer look, I've come to Switzerland's Sphinx Observatory,
0:48:12 > 0:48:17perched precariously on the Jungfrau mountain.
0:48:30 > 0:48:37At nearly 3,600 metres, it's one of the highest peaks in the Alps.
0:48:42 > 0:48:45And after the sun has sunk below the horizon...
0:48:47 > 0:48:51..it's a great place to gaze at the stars.
0:48:59 > 0:49:04Well, it's a really clear night, so you can see loads of stars.
0:49:04 > 0:49:07There's Sirius over here, the brightest star in the night sky
0:49:07 > 0:49:12and right here a really recognisable constellation, which is Orion.
0:49:12 > 0:49:15Have people always picked out Orion
0:49:15 > 0:49:17as a significant pattern in the night sky?
0:49:17 > 0:49:21It seems like different cultures all picked out that group
0:49:21 > 0:49:22as being a significant one.
0:49:22 > 0:49:25They all have different legends about it.
0:49:25 > 0:49:28The Egyptians associated it with Osiris, their god of death and rebirth
0:49:28 > 0:49:31Other cultures group them together.
0:49:31 > 0:49:34A native American tribe called the three stars of the belt,
0:49:34 > 0:49:36the three footprints of the flee god.
0:49:36 > 0:49:41One group of the Aborigines in Australia called it the canoe.
0:49:46 > 0:49:51Today, we don't need legends to explain the patterns in the stars
0:49:51 > 0:49:55because we know their precise positions in space.
0:49:59 > 0:50:02And we don't just know where they are now,
0:50:02 > 0:50:05we know where they were yesterday and where they'll be
0:50:05 > 0:50:08millions of years into the future.
0:50:09 > 0:50:14So the Sun and all the stars in our galaxy, including the stars in Orion,
0:50:14 > 0:50:17are all moving in orbits around the centre of the galaxy,
0:50:17 > 0:50:21but like a swarm of bees, although they're all moving in roughly the same direction,
0:50:21 > 0:50:26they all follow their own paths and that means that their positions will change,
0:50:26 > 0:50:28as thousands of years tick by.
0:50:28 > 0:50:32And now we're two-and-a-half million years in the future
0:50:32 > 0:50:36and the constellation of Orion has completely gone.
0:50:37 > 0:50:43In fact, thousands of years ago our ancestors would have seen different patterns in the sky
0:50:43 > 0:50:48and our descendants, millions of years in the future, will also see different patterns.
0:50:57 > 0:51:02The reason we can predict how the stars will move into the far future
0:51:02 > 0:51:05is because we've uncovered the rules that govern their behaviour.
0:51:07 > 0:51:11And we've found these rules not in the heavens, but in numbers.
0:51:18 > 0:51:23It's only through the Code that we can understand the laws that govern the universe.
0:51:48 > 0:51:51Laws that describe everything from the motion of the planets
0:51:51 > 0:51:54to the flight of projectile.
0:51:55 > 0:51:58When you watch the fireball fly through the air
0:51:58 > 0:52:01then it appears in the first part of its flight,
0:52:01 > 0:52:03when it's just left the trebuchet,
0:52:03 > 0:52:07that it's accelerating upwards and then it begins to slow down,
0:52:07 > 0:52:09before it stops just above me
0:52:09 > 0:52:13and then, finally, accelerates back down towards the ground.
0:52:18 > 0:52:20But if you analyse the flight using numbers,
0:52:20 > 0:52:23it reveals something rather surprising.
0:52:25 > 0:52:29When you plot a graph of the projectile's vertical speed
0:52:29 > 0:52:31against time...
0:52:32 > 0:52:35..you then you get a graph which looks like this.
0:52:40 > 0:52:43To start with, the projectile is moving upwards
0:52:43 > 0:52:46so it's vertical speed is positive, but decreasing.
0:52:48 > 0:52:52As it reaches the top of its arc, the vertical speed becomes negative
0:52:52 > 0:52:57as the fireball turns round and falls back to Earth.
0:53:00 > 0:53:04Because the graph is going like this, it means that the projectile,
0:53:04 > 0:53:09from the moment it leaves the trebuchet, is actually slowing down.
0:53:09 > 0:53:13So at no point during the flight is it ever accelerating upwards.
0:53:19 > 0:53:25Throughout its flight, the fireball is accelerating downwards
0:53:25 > 0:53:28towards the Earth at a constant rate.
0:53:29 > 0:53:32Something you would never realise simply by watching it
0:53:32 > 0:53:35fly through the air.
0:53:38 > 0:53:40And this is a profound truth
0:53:40 > 0:53:43about one of the fundamental forces of nature...
0:53:45 > 0:53:47..gravity.
0:53:48 > 0:53:51Drop, throw, fire or launch anything you like -
0:53:51 > 0:53:54a rock, a bullet, a ball or even a pot plant
0:53:54 > 0:53:57and it will accelerate towards the ground at a constant rate
0:53:57 > 0:54:01of 9.8 metres per second, per second.
0:54:01 > 0:54:05This is a fundamental law of gravity on our planet.
0:54:05 > 0:54:10But it's only revealed by changing the flight path of the object into numbers.
0:54:15 > 0:54:19Appreciating this simple fact about how gravity works on Earth
0:54:19 > 0:54:25is the first step towards understanding gravity everywhere.
0:54:38 > 0:54:44It's the foundation stone of Newton's Law of Universal Gravitation.
0:54:45 > 0:54:50A mathematical theory that can describe the orbits of the planets,
0:54:50 > 0:54:55predict the passage of the stars into the distant future...
0:54:57 > 0:55:03..and has even enabled human kind to step foot on the Moon.
0:55:07 > 0:55:13The laws that command the heavens are written in the Code.
0:55:24 > 0:55:28'We call them the door mats, the large ones.
0:55:28 > 0:55:31'Two-and-a-half million years in the future...
0:55:31 > 0:55:34'This isn't imaginery, this is real!
0:55:38 > 0:55:43'You don't need to know what that means to know that animal's not happy.
0:55:43 > 0:55:44'Whatever circle I take,
0:55:44 > 0:55:47'you're going to get a number which starts 3.14.'
0:55:51 > 0:55:56It's an incredible thought that the only way we can really make sense of our world
0:55:56 > 0:55:59is by using the abstract world of numbers.
0:55:59 > 0:56:04And yet those numbers have allowed us to take our first tentative steps off our planet.
0:56:04 > 0:56:09They've also given us the technology to transform our surroundings.
0:56:11 > 0:56:14'A hidden Code underpins the world around us.
0:56:17 > 0:56:20'A Code that has the power to unlock the rules that cover the universe.'
0:56:24 > 0:56:28This place was constructed to satisfy a spiritual need.
0:56:28 > 0:56:33But we couldn't have built it without the power of the Code.
0:56:33 > 0:56:38For me, it's an exquisite example of the beauty and potency of mathematics.
0:56:49 > 0:56:52From the patterns and numbers all around us,
0:56:52 > 0:56:56we've deciphered a hidden code.
0:57:09 > 0:57:14We've revealed a strange and intriguing numerical world,
0:57:14 > 0:57:15totally unlike our own.
0:57:17 > 0:57:23Yet it's a Code that also describes our world with astonishing accuracy.
0:57:29 > 0:57:33And has given us unprecedented power to describe...
0:57:36 > 0:57:38..control...
0:57:40 > 0:57:42..and predict our surroundings.
0:57:55 > 0:58:00The fact that the Code provides such a successful description of nature
0:58:00 > 0:58:03is for many one of the greatest mysteries of science.
0:58:04 > 0:58:07I think the only explanation that makes sense for me
0:58:07 > 0:58:10is that by discovering these connections,
0:58:10 > 0:58:13we have in fact uncovered some deep truth about the world.
0:58:13 > 0:58:17That perhaps, the Code is THE truth of the universe
0:58:17 > 0:58:21and it's numbers that dictate the way the world must be.
0:58:28 > 0:58:30Go to...
0:58:33 > 0:58:36..to find clues to help you solve the Code's treasure hunt.
0:58:36 > 0:58:39Plus, get a free set of mathematical puzzles and a treasure hunt clue
0:58:39 > 0:58:42when you follow the links to The Open University
0:58:42 > 0:58:45or call 0845 366 8026.
0:58:59 > 0:59:02Subtitles by Red Bee Media Ltd
0:59:02 > 0:59:05E-mail subtitling@bbc.co.uk