Numbers The Code


Numbers

Marcus du Sautoy reveals a hidden numerical code that underpins all nature. He reveals how significant numbers appear throughout the natural world and govern the planet.


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Transcript


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BOY: 'One for sorrow

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'Two for mirth

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GIRL: 'Three for a wedding

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'And four for death

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BOY: 'Nine for hell.'

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GIRL: '666.'

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Hidden within this cathedral are clues to a mystery,

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something that could help answer

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one of humanity's most enduring questions...

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..why is the world the way it is?

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The 13th-century masons who constructed this place

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had glimpsed a deep truth

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and they built a message into its very walls

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in the precise proportions of this magnificent cathedral.

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To the medieval clergy,

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these divine numbers were created by God.

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But to me, they're evidence of something else,

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a hidden code that underpins the world around us,

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a code that has the power to unlock the laws that govern the universe.

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As a mathematician, I'm fascinated by the numbers

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and patterns we see all around us...

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..numbers and patterns that connect everything

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from fish to circles

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and from our ancient past

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to the far future.

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INDISTINCT COMMENT

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Together they make up the Code...

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..an abstract world of numbers...

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..that has given us the most detailed description of our world we've ever had.

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For centuries, people have seen significant numbers everywhere...

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..an obsession that's left its mark in the stones of this medieval cathedral.

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In the 12th century, religious scholars here in Chartres

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became convinced these numbers were intrinsically linked to the divine...

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..an idea that dates back to the dawn of Christianity.

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The fourth-century Algerian cleric St Augustine believed

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that seven was so special that it represented the entire universe.

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He described how seven embraced all created things

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and ten was beyond even the universe

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because it was seven plus the three aspects of the Holy Trinity -

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Father, Son and Holy Ghost.

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12 was also hugely important, not simply because there are 12 tribes of Israel or 12 disciples of Jesus,

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but because 12 is divisible by one, two, three, four, six and 12 itself,

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more than any other number around it.

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For St Augustine, numbers had to come from God

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because they obey laws that no man can change.

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Around 800 years after St Augustine,

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the 12th-century Chartres School also recognised their significance.

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It's thought that, under their influence, sacred numbers

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were built into the structure of this majestic building.

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Numbers, they believed, held the key to the mystery of creation.

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I've spent my entire working life studying numbers,

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and for me they're more than just abstract entities.

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They describe the world around us.

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Although I don't share their religious beliefs, I can't help

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feeling something in common with the people who built this place.

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I share their awe and wonder at the beauty of numbers.

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For them, those numbers brought them closer to God, but I think they're important for another reason,

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because I believe they're the key to making sense of our world.

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Numbers have given us an unparalleled ability to understand our universe.

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And in places, this code literally emerges from the ground.

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Rural Alabama,

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spring 2011.

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Warm, lush and peaceful.

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But this year, there's a plague coming.

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While some locals are moving out,

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Dr John Cooley has driven thousands of miles to be here.

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He's on the trail of one of the area's strangest residents.

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We have been driving around looking for the emergences for about three and a half weeks.

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I've driven 7,200 miles since Good Friday trying to figure out where these things are.

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What makes these insects so remarkable is their bizarre lifecycle.

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For 12 whole years, they live hidden underground, in vast numbers.

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Then, in their 13th year...

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at precisely the same time...

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..they all burrow out from the earth to breed.

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At the full part of the emergence, there will be millions of insects out per acre. They'll be everywhere.

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It really is insect mayhem.

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This is the periodical cicada.

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This one is a male...

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..and you know that because on the abdomen,

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there's a pair of organs called timbles,

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and they're sound-producing organs.

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It's a little membrane that's vibrated, it makes a sound.

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Oh, yeah. I don't have to be frightened of these, do I?

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-No, no, they're absolutely harmless. They make wonderful pets.

-Really?

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-Mm-hm.

-They're quite ticklish.

-It's a harmless insect.

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It doesn't bite, it doesn't sting, nothing of that sort.

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Its only defence is safety in numbers.

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By emerging in such vast numbers, each individual cicada

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minimises its risk of being eaten.

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Because there are so many of them,

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their predators simply can't eat them fast enough.

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Well, you can certainly hear the cicadas.

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Yes, you can. There are probably millions of them up there.

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-Millions?

-Yeah, millions. What you probably don't realise is you're only hearing half the population.

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Only the males make these loud sounds.

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There are just as many females up there as well.

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And it's extraordinary to think that if we came here next year,

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-we wouldn't hear this sound at all?

-You'll have to come back in 13 years.

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So 2024 is when you'll hear the forest singing like this again?

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-That's right.

-That's amazing.

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Why have the cicadas evolved with this 13-year lifecycle as opposed to any other number?

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Well, you have to remember that these cicadas require large numbers to survive predators,

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and so we think that these long lifecycles in some way help them maintain large populations.

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John believes that, by appearing every 13 years,

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the cicadas minimise their chances of emerging at the same time

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as other cicadas with different lifecycles...

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..because if they were to interbreed, it could have disastrous consequences.

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The offspring would have unusual lifecycles.

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They're going to emerge a little bit here, a little bit there, some this year and some that year in small

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numbers, and that's key because if they emerge in small numbers, the predators eat them.

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The cicadas' survival depends on avoiding other broods.

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Imagine you've got a brood of cicadas that appears every six years.

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Now, let's suppose there's another brood

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which wants to try and avoid the red cicadas.

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One way to do that would be to appear less often in the forest, and that actually works.

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So let's suppose this brood appears every nine years.

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So if the green cicada appears every nine years,

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then it only coincides with the red cicada every 18 years.

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But, rather surprisingly, a smaller number, seven, works even better.

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Coming out every seven years instead of every nine

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means the cicadas appear together much less often.

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Now they only coincide every 42 years.

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That's just twice every century.

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And for the real cicadas,

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a 13-year lifecycle has exactly the same effect as seven does here

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because they both belong to a special series of numbers.

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Like 13, seven is a prime number.

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Unlike other numbers, primes can only be divided by themselves and one,

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and it's this property that means that numbers that are separated by primes

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are far less likely to coincide with multiples of other numbers.

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Because 13 is a prime number, a 13-year lifecycle

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makes the cicadas much less likely to coincide with other groups.

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Up in Georgia, there is another brood of periodical cicada

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and they, too, have a prime number lifecycle.

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They come out every 17 years.

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Because 13 and 17 are both prime numbers,

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the two broods only emerge together once every 221 years.

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Prime numbers are intimately linked to the cicadas' survival

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and, intriguingly,

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they're one of the most important elements of the Code,

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because the Code is a mathematical world,

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built from numbers.

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Just as atoms are the indivisible units that make up every physical object,

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so prime numbers are the indivisible building blocks of the Code.

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Prime numbers are indivisible, which means they can't be made

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by multiplying any other numbers together.

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But every non-prime number can be created by multiplying primes together.

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It's impossible to make any numbers without them.

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And if any primes are missing,

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there will always be some numbers you can't create.

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For me, the fact that the most fundamental units of mathematics

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can be found woven into the natural world

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is not only compelling evidence that the Code exists,

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but also that numbers underpin everything...

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..including our own biology.

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This is an innately human characteristic.

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Music is one of the things which defines who we are, and each culture has its own particular style.

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These guys make it seem so effortless, as if the notes

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are just thrown together, but that's simply an illusion.

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MUSIC ENDS, APPLAUSE

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Because, just as numbers govern the cicadas' lives,

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so they determine how WE hear sound.

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That's a C.

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And using this oscilloscope, I can get a picture of that note.

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So I can actually SEE the sound wave.

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Now, the height of the wave corresponds to how loudly I'm playing the note,

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so if I play the note very quietly...

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play it very loudly...I suddenly get a huge wave on the screen.

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The more important thing is the distance between the peaks of the wave,

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because that's determined by the pitch or frequency of the note.

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'The higher the note...

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'the shorter the distance between the peaks.'

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Now, look what happens when I play a C...

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..and compare that with the same note, a C, but an octave higher.

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Something rather surprising emerges,

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because now you can see that the higher note has twice

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as many peaks as the lower note,

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which means the frequency of the high C is twice that of the low C.

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And this happens whatever two notes you choose.

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Provided they're an octave apart, then their frequencies are going to be in this one-to-two ratio.

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Two notes which are an octave apart just sound nice together, and they're actually the most

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harmonious combination of notes that you can have.

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And that's because one to two is the simplest possible frequency relationship, and that's what

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music is all about, because it's these simple whole-number ratios that sound so good to the ear.

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A perfect fifth...

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is a frequency ratio of three to two.

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A perfect fourth...

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is four to three.

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And a slightly more complex sound, a minor sixth...

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..that's a frequency ratio of five to eight.

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Every combination of notes used in music is defined by simple ratios.

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Although we might not be aware of it, these numerical rules underpin

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everything from the simplest song to the most elaborate symphony.

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They're so deeply ingrained that when they're broken,

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we intuitively know something is wrong.

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Professor Judy Edworthy understands this more than most.

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She spends her time subjecting people to some of most unpleasant noises imaginable.

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Hi, Judy.

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-Ah, hello.

-Marcus.

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'Her research investigates the psychological effects of sound.

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'And by using complex ratios instead of simple ones, the noises she creates are nothing like music.'

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You can see just by looking at it it's not going to sound nice.

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The wave looks a mess.

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The wave is a mess. It's very difficult to see a pattern.

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CONSTANT DRONE

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OK. It sounds really quite odd now.

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It doesn't have any pitch. It sounds harsh and I could make it louder and that would make it harsher.

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When the various frequencies aren't simple multiples of one another,

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there's no common pattern for the ear to respond to,

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and the more complex you make the ratios, the more dissonant and harsh the sound will get.

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By monitoring her victims' reactions to these appalling noises,

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Professor Edworthy has found they have a very different effect

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on our minds than music.

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ALARM BEEPS

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HONKING

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WHIRRING

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-They're so unpleasant...

-HAMMERING

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..they shock our brains into action.

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For example, a siren.

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HIGH-PITCHED SIREN BLARES

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That's quite a harsh sound, but it's designed for a purpose - to get you out of the way.

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Sometimes you find these sounds in the animal world as well.

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So this, for example, this is a chimpanzee and an orang-utan.

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INTERMITTENT SCREECHING

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OK, these animals are obviously quite bothered by something.

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You don't need to know what that sound means to know that that animal's not happy

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and also that the other animals in that environment and us, for example, should just get out of the way.

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SHORT SCREECH

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So it's interesting that we really hear pattern,

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and when it isn't there, it creates an effect in all of us.

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LOW-PITCHED SCREECH

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Remarkably, it's numerical patterns in the Code

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that dictate the combinations of sounds we hear as music...

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RUSTLING

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-..and those we hear simply as noise.

-CHIRPING, SIREN

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BELL TOLLS

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And perhaps stranger still, it's these same numbers

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that are built into the walls of this medieval cathedral.

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Two notes which are an octave apart are going be in this one-to-two ratio.

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The width of the nave here is twice the distance between

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each of the columns that run up its length - a ratio of two to one.

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The most harmonious combination of notes from a pair.

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The altar divides the nave into a ratio of eight to five.

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A minor sixth...

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eight to five.

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A perfect fifth...

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three to two.

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A perfect fourth is four to three.

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Major third, five to four.

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And that's what music is all about.

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St Augustine believed these ratios were used by God to construct the universe

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and that that was why they produced harmony in music.

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By constructing their cathedral using the same ratios,

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the clergy at Chartres hoped to echo God's creation.

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This entire place is a symphony set in stone.

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Using the Code's numbers has created a building of awe-inspiring beauty.

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The only truth there is...

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Seemingly significant numbers...

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By searching for divine meaning in numbers,

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12th-century scholars had stumbled across elements of the Code.

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It's very difficult to see a pattern.

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Mysterious numbers and patterns that seem to be written into our biology.

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Its only defence is safety in numbers.

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And as we've looked closer, we haven't simply found more numbers -

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we've begun to uncover their strangest properties and started to see deep connections between them.

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Back in the distant past, in Neolithic times,

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around 4,000 years ago, an ancient people brought these stones here

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and arranged them like this.

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This is Sunkenkirk stone circle in Cumbria and it's one of around 1,000

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such structures that our ancient ancestors built across the UK.

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Stretching back into the mists of time,

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the circle has been steeped in mysticism.

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But whether the people who built this structure knew it or not,

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there is deep significance hidden inside this circle.

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OK, so I need to start by measuring the diameter

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of my circle, so that's the distance from one edge to the other.

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I need to go roughly through the centre.

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So that's 27 and 90.

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Right, so now I'm going to measure the circumference

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of the circle. So off we go.

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So around the outside.

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Oh, I've never got so much exercise doing maths before!

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And that's the circumference.

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So I've got 91 metres

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and 70 centimetres.

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I'm going to do a little calculation. I'm going to divide the circumference

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of the circle by the diameter.

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So 917 divided by 279.

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So that's roughly three...

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Bit of, er, mental arithmetic, not a mathematician's strongest point.

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OK, two lots of 279,

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so...

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not far out from what I was hoping for.

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So when I do that, I get roughly 3.2 as the answer.

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My measurements weren't very precise...

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..but my answer is close to a mysterious number hidden within every circle.

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So, for example, let's take this circular plate here.

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I'm going to measure its diameter.

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26.4 centimetres. Now its circumference.

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That's a bit trickier.

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82.9 centimetres.

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Divide the circumference by the diameter, I get 3.14.

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Now let's take another circle. Measure its diameter.

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12.8 centimetres.

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So the circumference is 40.2 centimetres.

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Divide the circumference by the diameter and I get 3.14.

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In fact, whatever circle I take, divide the circumference

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by the diameter and you're going to get a number which starts 3.14.

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This is a number we call pi.

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No matter where the circles are, no matter how big or small...

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..they will always contain pi.

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It's this universality of the number pi which tells you you've identified a piece of true Code.

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In fact, if you get another number,

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it means that you haven't got a circle.

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In some sense, pi is the essence of circleness,

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distilled into the language of the Code.

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And because circles and curves crop up again and again in nature,

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pi can be found all around us.

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It's in the gentle curve of a river...

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..the sweep of a coast line...

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..and the shifting patterns of the desert sands.

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Pi seems written into the structures and processes of our planet.

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But, strangely, pi also appears in places

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that seem to have nothing to do with circles.

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I started fishing Brighton in 1972.

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I've been a fisherman 40 years, catching Dover sole.

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That's the main target species for the English Channel.

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How many fish do you think you get a day?

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300 some days, 150 other days,

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so I'd say 200 would be average.

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And you've got me some Dover sole today so I can have a weigh of what you've caught today.

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-Yeah, you can play with them!

-OK!

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What's remarkable is that, with just a small amount of information...

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It's 180 grams.

0:30:050:30:07

..and by weighing a few fish...

0:30:080:30:10

That's a whopper.

0:30:100:30:12

..I can use the Code

0:30:120:30:13

to tell me things about not just today's catch...

0:30:130:30:16

360 grams. 50 grams. 110 grams.

0:30:160:30:20

..but about all the Dover sole Sam's ever fished...

0:30:200:30:24

Whoa, jeez, come back!

0:30:240:30:26

..I can even get an estimate for the largest sole

0:30:260:30:29

that Sam is likely to have caught during his career.

0:30:290:30:32

Right...

0:30:320:30:33

First , I need to work out what the average weight of a fish is,

0:30:330:30:37

or the mean,

0:30:370:30:40

so 140 plus 190

0:30:400:30:44

plus 150...

0:30:440:30:46

So now I need to work out the standard deviation, so that's 140 minus square that...

0:30:460:30:51

Bear with me, all right? Almost there.

0:30:510:30:54

So he said he fished for 40 years,

0:30:540:30:59

and eight weeks during the year, six days out of the week

0:30:590:31:04

and 200 sole each day,

0:31:040:31:09

so that gives you a total of 384,000 fish.

0:31:090:31:13

Using these numbers, I can calculate that the largest one

0:31:140:31:18

out of those 384,000 fish

0:31:180:31:21

should be about 1.3 kilograms, which is roughly three pounds.

0:31:210:31:26

So what's the largest Dover sole that you've caught in your career?

0:31:280:31:33

We call them door mats, the large ones,

0:31:330:31:36

and you maybe get four or five a season.

0:31:360:31:39

The largest, I'd say, was three to three and a half pounds.

0:31:390:31:44

An average Dover Sole is that sort of size

0:31:440:31:48

and these...

0:31:480:31:50

Wow, that's huge! Yeah!

0:31:500:31:53

It's a whopper. It's always nice to catch big stuff, you know.

0:31:530:31:57

Well, I think it is anyway. HE CHUCKLES

0:31:570:31:59

Using the Code, it's possible to estimate the size

0:32:040:32:07

of the biggest fish Sam's ever caught,

0:32:070:32:10

despite not weighing a single fish anywhere near that size.

0:32:100:32:15

Now, the reason this calculation is possible is because the distribution of the weights of fish,

0:32:190:32:26

in fact the distribution of lots of things like the height of people in the UK or IQ,

0:32:260:32:31

is given by this formula.

0:32:310:32:34

'This is the normal distribution equation,

0:32:340:32:37

'one of the most important bits of mathematics

0:32:370:32:41

'for understanding variation in the natural world.'

0:32:410:32:45

The most remarkable thing about this formula isn't so much what it does

0:32:450:32:50

as this term here, pi.

0:32:500:32:52

It seems totally bizarre

0:32:520:32:54

that a bit of the Code that has something to do with the geometry of a circle

0:32:540:32:58

can help you to calculate the weight of fish.

0:32:580:33:01

Pi shouldn't have anything to do with fish, yet there it is.

0:33:010:33:06

Just as the circle appears everywhere in nature,

0:33:140:33:18

so pi crops up again and again in the mathematical world.

0:33:180:33:23

It's an astonishing example of the interconnectedness of the Code.

0:33:240:33:30

A glimpse into a world where numbers don't just have strange connections,

0:33:300:33:35

they have deeply puzzling properties of their own.

0:33:350:33:39

Pi is what's known as an irrational number.

0:33:420:33:45

Written as a decimal, it has an infinite number of digits

0:33:470:33:51

arranged in a sequence that never repeats.

0:33:510:33:55

And it's thought that any number you can possibly imagine

0:33:570:34:01

will appear in pi somewhere, from my birthday

0:34:010:34:06

to the answer to life, the universe and everything.

0:34:060:34:10

Because they go on for ever, we can never know all the digits

0:34:130:34:16

that make up pi.

0:34:160:34:17

But, luckily, we only need the first 39

0:34:170:34:21

to calculate the circumference of a circle the size of the entire observable universe,

0:34:210:34:26

accurate to the radius of a single hydrogen atom.

0:34:260:34:30

But as strange as Pi is, it does at least describe a physical object.

0:34:370:34:42

Some numbers don't make any sense in real world,

0:34:430:34:47

despite the fact we use them all the time.

0:34:470:34:50

Numbers, like negative numbers.

0:34:500:34:53

It's impossible to trade anything, stocks, shares, currency,

0:34:550:35:00

even fish, without negative numbers.

0:35:000:35:03

Most of us are comfortable them.

0:35:030:35:05

Even though we may not like it, we understand what it means

0:35:050:35:08

to have a negative bank balance.

0:35:080:35:10

But when you start to think about it,

0:35:100:35:12

there's something deeply strange about negative numbers,

0:35:120:35:16

cos they don't seem to correspond to anything real at all.

0:35:160:35:20

The deeper we look into the Code, the more bizarre it becomes.

0:35:220:35:27

It's easy to imagine one fish or two fish, or no fish at all.

0:35:320:35:39

It's much harder to imagine what minus-one fish looks like.

0:35:390:35:43

Negative numbers are so odd that if I have minus-one fish and you give me a fish,

0:35:430:35:48

then all you can be certain of is that I've got no fish at all.

0:35:480:35:52

Numbers, can exist regardless of whether they make any sense in the physical world.

0:35:590:36:05

And if you think that's odd, some numbers are so strange

0:36:090:36:14

they don't even seem to make sense as numbers.

0:36:140:36:17

Now, this is one of the most basic facts of mathematics.

0:36:180:36:22

A positive number multiplied by another positive number is a positive number.

0:36:220:36:27

So for example, one times one is one.

0:36:270:36:33

A negative number multiplied by another negative number

0:36:330:36:37

also gives a positive number.

0:36:370:36:40

So for example, minus-one times minus-one is plus-one.

0:36:400:36:46

'It's not only a rule, it's a proven truth of multiplication.

0:36:460:36:52

'Whenever the signs are the same, the product is always positive.'

0:36:520:36:56

From this, it's obvious

0:36:560:36:58

if I take any number and multiply it by itself,

0:36:580:37:00

then the answer is going to be positive.

0:37:000:37:03

However, in the Code,

0:37:030:37:05

there's a special number which breaks this rule.

0:37:050:37:08

When I multiply it by itself, it gives the answer minus-one.

0:37:080:37:12

It's impossible to imagine what this number could be,

0:37:120:37:16

because there simply is no number

0:37:160:37:19

that when multiplied by itself, gives minus-one.

0:37:190:37:23

This isn't a number I can calculate. I can't show you this number.

0:37:230:37:28

Nevertheless, we've given this number a name.

0:37:280:37:30

It's called "i", and it's part of a whole class of new numbers

0:37:300:37:33

called imaginary numbers.

0:37:330:37:35

Calculating with imaginary numbers is the mathematical equivalent

0:37:370:37:41

of believing in fairies.

0:37:410:37:43

But even these strangest elements of the Code turn out to have

0:37:450:37:49

some very practical applications.

0:37:490:37:52

The ground's close, will you call me, please, 1-1-9 next...

0:37:560:38:00

Runway 25, clear to land. Surface is 1-3-0, less than five minutes.

0:38:030:38:08

'Especially on a day like this.'

0:38:080:38:11

8-5 Foxtrot, thank you, vacate next right and park yourself 1-3 short.

0:38:140:38:19

'8-5 Foxtrot, 8-2-0, both making approach down direct and right, 2-5.'

0:38:190:38:24

So where's this one coming from?

0:38:240:38:26

That is from Barcelona. It's an Easyjet flight, EZZ6402.

0:38:260:38:30

Don't know how many people are on board, but it seats about 190.

0:38:300:38:34

And here he is. He's getting pretty close now.

0:38:340:38:36

Just less than two miles till he lands.

0:38:360:38:38

What information is the radar giving you about the aeroplanes?

0:38:380:38:42

The first and most important thing is the position of the aircraft.

0:38:420:38:46

The yellow slash there is where the aircraft is.

0:38:460:38:49

You've got the blue trail, the history of where the aircraft's been.

0:38:490:38:53

From that you get two things - you get its rough heading, where he's going, and its speed.

0:38:530:38:58

The longer the trail, the faster the aircraft's going.

0:38:580:39:00

Radar works by sending out a pulse of radio waves

0:39:060:39:09

and analysing the small fraction of the signal that's reflected back.

0:39:090:39:13

Complex computation is then needed to distinguish moving objects,

0:39:170:39:21

like planes, from the stationary background.

0:39:210:39:25

RADIO COMMUNICATION

0:39:250:39:28

At the heart of that analysis lies "i", the number that cannot exist.

0:39:280:39:35

Imaginary numbers are useful for working out the complex way

0:39:360:39:41

radio waves interact with each other.

0:39:410:39:43

It seems to be the right language to describe their behaviour.

0:39:430:39:47

Now, you could do these calculations with ordinary numbers.

0:39:470:39:50

But they're so cumbersome,

0:39:500:39:52

by the time you've done the calculation the plane's moved to somewhere else.

0:39:520:39:56

Attitude 6,000 on a squawk of 7-7-1-5.

0:39:560:40:00

Using imaginary numbers makes the calculation simpler

0:40:000:40:03

that you can track the planes in real time.

0:40:030:40:06

In fact without them, radar would be next to useless for Air Traffic Control.

0:40:060:40:11

It's kind of amazing that this abstract idea lands planes.

0:40:150:40:19

It's a bit surprising, you're talking about imaginary numbers

0:40:190:40:22

-and this isn't imaginary, this is real.

-This is very real.

0:40:220:40:24

I'm surprised at the fact that something so abstract

0:40:240:40:28

is being used in such a concrete way.

0:40:280:40:30

As strange as it may seem, the code provides us

0:40:450:40:48

with an astonishingly successful description of our world.

0:40:480:40:52

Its most ethereal numbers have starkly real applications.

0:40:580:41:03

Its patterns can explain one of the most profound processes in nature -

0:41:030:41:09

how living things grow.

0:41:090:41:12

This is a picture of something I've been fascinated by

0:41:150:41:18

ever since I became a mathematician.

0:41:180:41:20

It's an X-ray of a marine animal called a nautilus.

0:41:200:41:25

And this spiral here is one of the iconic images of mathematics.

0:41:250:41:30

Now, while I've seen pictures like this hundreds of times,

0:41:300:41:33

I've never actually seen the animal for real.

0:41:330:41:36

'At Brooklyn College, biologist Jennifer Basil keeps five of these aquatic denizens,

0:41:390:41:44

'for her research into the evolution of intelligence.'

0:41:440:41:48

We keep the animals in these tall tanks because they're naturally active at night

0:41:500:41:54

and they like darkness, they live in deep water.

0:41:540:41:58

They also like to go up and down in the water column,

0:41:580:42:00

-that kind of makes them happy.

-OK!

0:42:000:42:02

-We give them the five-star treatment here.

-Right...

0:42:020:42:05

-This is Number Five.

-Ah, wow.

-Yeah.

0:42:070:42:10

Gosh, big eyes.

0:42:100:42:11

-They have huge eyes, great for seeing in low light conditions.

-Right.

0:42:110:42:15

-So, here's that beautiful shell.

-Yeah.

0:42:160:42:18

And the striping pattern helps them hide where they live.

0:42:180:42:22

I've never seen the animal before inside the shell, what is it?

0:42:380:42:43

They're related to octopuses, squids and cuttlefish.

0:42:430:42:46

It's a little bit like an octopus with a shell

0:42:460:42:49

and what's amazing about them is that their lineage

0:42:490:42:53

is hundreds of millions of years old and they haven't changed very much

0:42:530:42:57

in all that time. We call them a living fossil.

0:42:570:42:59

It's a great opportunity to look at an ancient brain and behaviour

0:42:590:43:04

and they're a wonderful way to study the evolution of intelligence.

0:43:040:43:07

So are these guys intelligent, then?

0:43:070:43:10

Some are smarter than others, like that's Number Four,

0:43:100:43:14

he outperforms everybody in all the memory tests.

0:43:140:43:17

He's quite active all the time, he's quite engaging.

0:43:170:43:20

If you put your in the water he comes up to you,

0:43:200:43:22

whereas Number Three, who happens to be a teenager,

0:43:220:43:26

is I'd guess you'd say more shy and you put him in a new place

0:43:260:43:29

and he sort of just attaches to the wall and sits there.

0:43:290:43:33

I'm interested in the shell as a mathematician,

0:43:330:43:36

but what does the nautilus use the shell for?

0:43:360:43:38

I think the most obvious use is protection.

0:43:380:43:41

They also use it for buoyancy.

0:43:420:43:44

They only live in the front chamber

0:43:440:43:46

and all the other chambers are filled with gas

0:43:460:43:49

and with some fluid.

0:43:490:43:50

By regulating that, they can gently and passively move up and down

0:43:500:43:55

in the water like a submarine.

0:43:550:43:57

The really cool thing they can do

0:43:570:43:59

is they can actually survive on the oxygen in the chambers,

0:43:590:44:03

if there's a period where the oxygen goes down in the oceans.

0:44:030:44:07

It's one of the reasons why they've lived for millions of years.

0:44:070:44:11

It's a really great adaptation. The shell is really amazing.

0:44:110:44:14

But perhaps even more remarkably, the rules this ancient creature

0:44:170:44:21

uses to construct its home

0:44:210:44:23

are written in the language of the Code.

0:44:230:44:27

HORNS BLARE

0:44:270:44:29

The nautilus shell is one of the most beautiful and intricate structures in nature.

0:44:360:44:41

Here you can see the chambers. This is the one where it lives

0:44:410:44:44

and these are the ones it uses for buoyancy.

0:44:440:44:47

Now, at first sight, this looks like a really complex shape,

0:44:470:44:50

but if I measure the dimensions of these chambers

0:44:500:44:53

a clear pattern begins to emerge.

0:44:530:44:56

Now there doesn't seem to be any connection between these numbers,

0:45:090:45:13

but look what happens when I take each number

0:45:130:45:16

and divide it by the previous measurement.

0:45:160:45:19

If I take 3.32 and divide by 3.07,

0:45:190:45:25

I get 1.08.

0:45:250:45:27

Divide 3.59 by 3.32

0:45:270:45:30

and I get 1.08.

0:45:300:45:33

Take 3.88 and divide by 3.59 and I get, again, 1.08.

0:45:330:45:37

So every time I do this calculation, I get the same number.

0:45:390:45:44

So although it's not clear by looking at the shell,

0:45:440:45:46

this tells us that the nautilus is growing at a constant rate.

0:45:460:45:51

Everytime the nautilus builds a new room, the dimensions of that room

0:45:510:45:55

are 1.08 times the dimensions of the previous one.

0:45:550:45:58

And it's just by following this simple mathematical rule

0:45:580:46:02

that the nautilus builds this elegant spiral.

0:46:020:46:05

And because many living things grow in a similar way,

0:46:080:46:12

these spirals are everywhere.

0:46:120:46:15

The rules nature uses to create its patterns are found in the Code.

0:46:170:46:22

Behind the world we inhabit, there's a strange and wonderful mathematical realm.

0:46:500:46:55

They're actually related to octopus, squids and cuttlefish.

0:46:550:46:59

They're quite ticklish.

0:46:590:47:00

The numbers and connections at its heart describe the processes we see all around us.

0:47:040:47:10

Bear with me, all right?

0:47:100:47:11

But the Code doesn't just contain the rules that govern our planet -

0:47:160:47:21

its numbers also describe the laws that control the entire universe.

0:47:210:47:27

For centuries, we've gazed out into the night's sky

0:47:390:47:44

and tried to make sense of the patterns we see in the stars.

0:47:440:47:49

To take a closer look, I've come to Switzerland's Sphinx Observatory,

0:48:070:48:12

perched precariously on the Jungfrau mountain.

0:48:120:48:17

At nearly 3,600 metres, it's one of the highest peaks in the Alps.

0:48:300:48:37

And after the sun has sunk below the horizon...

0:48:420:48:45

..it's a great place to gaze at the stars.

0:48:470:48:51

Well, it's a really clear night, so you can see loads of stars.

0:48:590:49:04

There's Sirius over here, the brightest star in the night sky

0:49:040:49:07

and right here a really recognisable constellation, which is Orion.

0:49:070:49:12

Have people always picked out Orion

0:49:120:49:15

as a significant pattern in the night sky?

0:49:150:49:17

It seems like different cultures all picked out that group

0:49:170:49:21

as being a significant one.

0:49:210:49:22

They all have different legends about it.

0:49:220:49:25

The Egyptians associated it with Osiris, their god of death and rebirth

0:49:250:49:28

Other cultures group them together.

0:49:280:49:31

A native American tribe called the three stars of the belt,

0:49:310:49:34

the three footprints of the flee god.

0:49:340:49:36

One group of the Aborigines in Australia called it the canoe.

0:49:360:49:41

Today, we don't need legends to explain the patterns in the stars

0:49:460:49:51

because we know their precise positions in space.

0:49:510:49:55

And we don't just know where they are now,

0:49:590:50:02

we know where they were yesterday and where they'll be

0:50:020:50:05

millions of years into the future.

0:50:050:50:08

So the Sun and all the stars in our galaxy, including the stars in Orion,

0:50:090:50:14

are all moving in orbits around the centre of the galaxy,

0:50:140:50:17

but like a swarm of bees, although they're all moving in roughly the same direction,

0:50:170:50:21

they all follow their own paths and that means that their positions will change,

0:50:210:50:26

as thousands of years tick by.

0:50:260:50:28

And now we're two-and-a-half million years in the future

0:50:280:50:32

and the constellation of Orion has completely gone.

0:50:320:50:36

In fact, thousands of years ago our ancestors would have seen different patterns in the sky

0:50:370:50:43

and our descendants, millions of years in the future, will also see different patterns.

0:50:430:50:48

The reason we can predict how the stars will move into the far future

0:50:570:51:02

is because we've uncovered the rules that govern their behaviour.

0:51:020:51:05

And we've found these rules not in the heavens, but in numbers.

0:51:070:51:11

It's only through the Code that we can understand the laws that govern the universe.

0:51:180:51:23

Laws that describe everything from the motion of the planets

0:51:480:51:51

to the flight of projectile.

0:51:510:51:54

When you watch the fireball fly through the air

0:51:550:51:58

then it appears in the first part of its flight,

0:51:580:52:01

when it's just left the trebuchet,

0:52:010:52:03

that it's accelerating upwards and then it begins to slow down,

0:52:030:52:07

before it stops just above me

0:52:070:52:09

and then, finally, accelerates back down towards the ground.

0:52:090:52:13

But if you analyse the flight using numbers,

0:52:180:52:20

it reveals something rather surprising.

0:52:200:52:23

When you plot a graph of the projectile's vertical speed

0:52:250:52:29

against time...

0:52:290:52:31

..you then you get a graph which looks like this.

0:52:320:52:35

To start with, the projectile is moving upwards

0:52:400:52:43

so it's vertical speed is positive, but decreasing.

0:52:430:52:46

As it reaches the top of its arc, the vertical speed becomes negative

0:52:480:52:52

as the fireball turns round and falls back to Earth.

0:52:520:52:57

Because the graph is going like this, it means that the projectile,

0:53:000:53:04

from the moment it leaves the trebuchet, is actually slowing down.

0:53:040:53:09

So at no point during the flight is it ever accelerating upwards.

0:53:090:53:13

Throughout its flight, the fireball is accelerating downwards

0:53:190:53:25

towards the Earth at a constant rate.

0:53:250:53:28

Something you would never realise simply by watching it

0:53:290:53:32

fly through the air.

0:53:320:53:35

And this is a profound truth

0:53:380:53:40

about one of the fundamental forces of nature...

0:53:400:53:43

..gravity.

0:53:450:53:47

Drop, throw, fire or launch anything you like -

0:53:480:53:51

a rock, a bullet, a ball or even a pot plant

0:53:510:53:54

and it will accelerate towards the ground at a constant rate

0:53:540:53:57

of 9.8 metres per second, per second.

0:53:570:54:01

This is a fundamental law of gravity on our planet.

0:54:010:54:05

But it's only revealed by changing the flight path of the object into numbers.

0:54:050:54:10

Appreciating this simple fact about how gravity works on Earth

0:54:150:54:19

is the first step towards understanding gravity everywhere.

0:54:190:54:25

It's the foundation stone of Newton's Law of Universal Gravitation.

0:54:380:54:44

A mathematical theory that can describe the orbits of the planets,

0:54:450:54:50

predict the passage of the stars into the distant future...

0:54:500:54:55

..and has even enabled human kind to step foot on the Moon.

0:54:570:55:03

The laws that command the heavens are written in the Code.

0:55:070:55:13

'We call them the door mats, the large ones.

0:55:240:55:28

'Two-and-a-half million years in the future...

0:55:280:55:31

'This isn't imaginery, this is real!

0:55:310:55:34

'You don't need to know what that means to know that animal's not happy.

0:55:380:55:43

'Whatever circle I take,

0:55:430:55:44

'you're going to get a number which starts 3.14.'

0:55:440:55:47

It's an incredible thought that the only way we can really make sense of our world

0:55:510:55:56

is by using the abstract world of numbers.

0:55:560:55:59

And yet those numbers have allowed us to take our first tentative steps off our planet.

0:55:590:56:04

They've also given us the technology to transform our surroundings.

0:56:040:56:09

'A hidden Code underpins the world around us.

0:56:110:56:14

'A Code that has the power to unlock the rules that cover the universe.'

0:56:170:56:20

This place was constructed to satisfy a spiritual need.

0:56:240:56:28

But we couldn't have built it without the power of the Code.

0:56:280:56:33

For me, it's an exquisite example of the beauty and potency of mathematics.

0:56:330:56:38

From the patterns and numbers all around us,

0:56:490:56:52

we've deciphered a hidden code.

0:56:520:56:56

We've revealed a strange and intriguing numerical world,

0:57:090:57:14

totally unlike our own.

0:57:140:57:15

Yet it's a Code that also describes our world with astonishing accuracy.

0:57:170:57:23

And has given us unprecedented power to describe...

0:57:290:57:33

..control...

0:57:360:57:38

..and predict our surroundings.

0:57:400:57:42

The fact that the Code provides such a successful description of nature

0:57:550:58:00

is for many one of the greatest mysteries of science.

0:58:000:58:03

I think the only explanation that makes sense for me

0:58:040:58:07

is that by discovering these connections,

0:58:070:58:10

we have in fact uncovered some deep truth about the world.

0:58:100:58:13

That perhaps, the Code is THE truth of the universe

0:58:130:58:17

and it's numbers that dictate the way the world must be.

0:58:170:58:21

Go to...

0:58:280:58:30

..to find clues to help you solve the Code's treasure hunt.

0:58:330:58:36

Plus, get a free set of mathematical puzzles and a treasure hunt clue

0:58:360:58:39

when you follow the links to The Open University

0:58:390:58:42

or call 0845 366 8026.

0:58:420:58:45

Subtitles by Red Bee Media Ltd

0:58:590:59:02

E-mail [email protected]

0:59:020:59:05

Marcus du Sautoy reveals a hidden numerical code that underpins all nature. A code that has the power to explain everything, from the numbers and shapes we see all around us to the rules that govern our own lives. In this first episode, Marcus reveals how significant numbers apear throughout the natural world. They're part of a hidden mathematical world that contains the rules that govern everything on our planet and beyond.


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