Shapes The Code

OVERLAPPING VOICES

This is the Giant's Causeway at the northern tip of Northern Ireland,

and it's famed for these strange angular rocks.

There are 40,000 of them crammed into this small area of coastline.

What makes them so striking is that they're so regular, so simple,

they just don't seem to fit in to this rugged natural environment.

The mystery of these hexagonal rock formations has inspired

storytellers and composers.

But their strange beauty is only the start of the story.

Because these stones tell of a hidden geometric force

that underpins and pervades all nature.

And if we can uncover that force,

it'll help us to explain the shape of everything...

from the smallest microbe, to the construction of these stones

and the formation of the world itself.

As a mathematician, I'm fascinated by the numbers

and shapes we see all around us...

..connecting everything, from bees

to bubbles

and the handwork of our distant ancestors

to the imagination of our greatest modern artists.

These are the hidden connections that make up the Code...

..an abstract, enigmatic world of numbers that has given us

the most detailed description of our world we've ever had.

Ever since they settled here, over 30,000 years ago,

people have tried to explain these remarkable hexagonal columns

poking out of the Irish Sea.

Why are they the shape they are?

And where did they come from in the first place?

Legend has it that this peninsula was once home to a giant

called Fionn mac Cumhaill.

One day the giant got into an argument with another giant called Benandonner

who lived 80 miles away across the sea in Scotland.

The giants hurled insults at each other,

swiftly followed by a few stones.

And things soon got out of hand.

Benandonner swore that if he was a better swimmer,

he'd come straight over to sort Fionn out.

Fionn was so enraged that he started picking up huge clumps of earth

and throwing them across the sea

so he could create a pathway for the Scottish giant to come and face him.

And that, legend has it, is what I'm standing on now.

The handiwork of a giant.

It's a nice story, but the reality is even more extraordinary.

Because what's written into these rocks is a fundamental truth

about the universe.

A truth that we can find written throughout the natural world.

These orchards in California,

are the site of one of the largest animal migrations on the planet.

Every spring, billions of bees are brought here

to help pollinate the almond trees.

Several thousand of these hives belong to Steve Godling.

-You go ahead and smoke it when we get it open.

-Yep.

-Right there.

-That's good.

Got this glued together very tight.

You want to try to get an outside one so as not to kill the queen.

You don't want to kill any of them but you particularly don't want to kill her.

-If you kill the queen, you've killed the hive.

-Wow!

That's one of the wonders of the natural world.

It's beautiful.

'The bees' honeycomb is a marvel of natural engineering.'

They've got plenty of honey.

'Everything they need is here.

'It's a place to raise their young and store their food.

'And it's all made from wax,

'a substance so labour intensive that the bees have to fly the equivalent

'of 12 times round the Earth to produce a single pound of it.'

-This almost looks man-made, manufactured.

-Yeah.

It doesn't look like something from the natural world.

-The precision, the fine straight lines that they've created is extraordinary.

-Right.

It's an engineering wonder, for sure.

-Look at the... It's perfect hexagons here.

-Yeah. It's amazing.

And, er, the hexagon is a very strong structure.

'The bees have made an identical pattern to the columns

'on the Giant's Causeway.

'Each cell is exactly like the others -

'six walls meeting precisely at 120 degrees.

'And every honeybee, everywhere in the world,

'knows how to build these shapes.

'It's as if the hexagon is built into the bee's DNA.'

You can see the bees going down inside the cell.

-It's almost exactly the same size as their bodies.

-Right.

Are they using their body like a ruler in some sense, to do the geometry?

That's an accurate description.

I know different races have a smaller body

and the cell size in their comb is smaller.

And each of the hexagons, how do they actually make a hexagon rather than some irregular shape?

They've just done it for thousands of years.

They were born to do it, they just instinctively know

that this is the shape of their home.

But there's more to the bees' behaviour than raw instinct.

There's another reason why they build in hexagons.

And to reveal that reason,

we need to turn to the universal language of all nature.

Mathematics.

The bees' primary need is to store as much honey as they can

while using as little precious wax as possible.

The bees' honeycomb is an amazing piece of engineering,

but why have they evolved to produce this hexagonal pattern?

They don't have too many choices.

If you try to put pentagons together, for example, they just don't fit together nicely.

Or circles leave lots of little gaps.

If they want to produce a network of regular shapes which fit together neatly

then you've really only got three options.

You can do equilateral triangles, or you could do squares,

or you can do the bees' hexagons.

But why of those three does the bee choose the hexagons?

Well, it turns out that the triangles actually use

much more wax than any of the other shapes.

Squares are a little better, but it's the hexagons which use the least amount of wax.

'It's a solution that was only mathematically proven a few years ago.

'The hexagonal array IS the most efficient storage solution

'the bees could have chosen.

'Yet with a little help from evolution,

'they worked it out for themselves millions of years ago.'

This is nature's Code at work,

and the bees are in tune with it.

It's easy to see why efficiency is important to the bees.

After all, it's hard work making wax.

But what could be the reason for the same pattern

being permanently engraved in the rock of the Giant's Causeway?

The geological processes that form these columns took place over thousands of years.

But to understand what happened, we need to look at structures that last for only a few seconds.

Soap films are mostly thinner than wavelengths of light.

About 20,000 times thinner than a human hair.

They're almost not here.

Probably the thinnest thing you've ever looked at

and got information back from was a soap film.

Tom Noddy is one of the world's foremost exponents of bubble art.

The different colours on a bubble are different thickness of soap film.

So looking over the colours of a bubble,

you're actually looking at a contour map of the surface of the bubble.

Bang.

So, like everything in nature, bubbles are just trying to economise,

they're trying to get as small as they possibly can.

But in the case of bubbles, they can do it perfectly.

A single bubble in the air is always a sphere.

At first sight, it seems obvious that the bubble should be round.

But why is the sphere so special?

The sphere is one surface, no corners, infinitely symmetrical.

Of all the shapes this bubble could be,

the sphere is the one with the smallest surface area,

which makes it the most efficient shape possible.

And it is because nature loves to use her resources effectively

that we can see spheres everywhere we look.

The Earth is round

because gravity pulls the planet's bulk into a ball around its core.

Water forms into spherical droplets -

the shape minimises the amount of surface tension needed to hold the droplet together.

And a spherical shape gives simple life forms,

like this Volvox plankton,

optimal contact with their surrounding environment.

But not everything is spherical.

And because bubbles are so thin and flexible

?we can use them to create other shapes.

So, a single bubble in the air is always a sphere.

But if they touch each other, they can save material for both of them by sharing a common wall.

And so they do.

If they can save surface area by taking advantage of their environment, they will.

So when you've got just one bubble, the sphere is the most efficient shape.

But as we add more bubbles, we see the geometry changing.

So, in this case,

we've got four bubbles and you can see them meeting at a point.

But put a shape in the middle, we don't get a spherical bubble,

we get, in fact, a little tetrahedron.

With four faces, they're not exactly flat, they're parts of spheres,

but each time, the bubbles are trying to find

the most efficient shape for the arrangement of bubbles.

So now we've got six bubbles, we've got a little cube appearing in the middle.

This is nature's laws at work.

The universe is always trying to find the most efficient solution it can.

And as we pop them, the bubbles change,

finding the most efficient, until we're left with a sphere again.

It has no choice.

But what's most remarkable is that those solutions

are so often neat, geometric shapes.

Wow!

That's a dodecahedron. That's fantastic.

And they're almost perfect pentagons. That's really surprising.

-They're not bulging really very much at all.

-That's right.

So, 12 bubbles around make 12 faces

and the most economical shape that they can make,

-the lowest energy, is the dodecahedron.

-Yeah.

The soap bubble reveals something fundamental about nature. It's lazy.

It tries to find the most efficient shape,

the one using the least energy, the least amount of space.

And it appears there ARE fixed rules about how it finds

these economic solutions.

The bubbles are incredibly dynamic, but each time one pops,

they're always trying to assume the most efficient shape,

the one that uses the least energy.

And what they're doing is trying to minimise the surface area

across the whole bubble structure.

This beautifully illustrates one of the fundamental rules of bubbles,

which is, three walls of a bubble will meet always at 120 degree angle.

Wherever you are in the foam, it's the same law.

But if we, in fact, made each of the bubbles the same size,

a rather magical shape starts to appear.

The hexagon.

'And when you pack lots of hexagons together,

'the pattern that spontaneously emerges is the familiar sight

'of a tightly ordered honeycomb.'

So when we see that pattern at the heart of the beehive,

it's echoing some of the fundamental geometrical rules of the universe.

It's the principles we see in bubbles that help explain where all structure comes from.

And it's those same fundamental laws of shape that played out

on the Giant's Causeway in the distant geological past.

50 million years ago, before there was any thought of warring giants,

this area was very unstable.

There was a huge amount of volcanic activity.

The molten rock forces its way through the chalk bed beneath my feet

and then spread out, forming a huge lava lake.

As it cooled, the lake contracted, and as it shrunk, it cracked.

And as the cracks spread, they sought out the most efficient path

through the lava,

which turned out to be this neat hexagonal pattern...

..leaving this monument to the order and economy of nature.

'It's an engineering wonder, for sure.'

The Code reveals itself where you would least expect it.

It defines the shape of honeycomb.

'They've just done it for thousands of years. They were born to do it.'

And it forms Ulster's epic coastline.

'..they just don't seem to fit in to this rugged natural environment.'

'Fionn mac Cumhaill.'

And it appears in the lazy efficiency of a soap film.

'About 20,000 times thinner than a human hair.'

These natural codes are so fundamental

that they've been appropriated by artists and architects to shape the modern world.

CHEERING

So this is the Olympic stadium that was built in Munich in 1972,

also scene of a rather famous victory for England.

A rare one, 5-1 to us against Germany.

It's really stunning

but I'm quite surprised at how insubstantial it feels.

It feels as though it could blow away in the wind.

It's got those features you expect in nature,

very elegant, but rather delicate feel to it.

So it's almost more like a cobweb than a man-made structure.

In 1972, which you have to remember is pre the computer age,

it was very difficult to build structures like this.

The distribution of forces that's going on inside this roof

are incredibly complicated.

It would be almost impossible to calculate by hand a shape like this

that would be both stable and affordable.

But the revolutionary engineer Frei Otto realised

that you don't have to do these calculations by hand.

Otto was desperate to find new shapes and forms to build,

so he looked to nature,

and the fundamental principles of the Code, for inspiration.

What Otto did was to make models like this one here.

It's constructed out of string, wires and these poles.

It doesn't look like much

but when I dip the string inside the soap solution and pull it up,

something rather surprising happens.

You start to see these beautiful shapes beginning to emerge

inside a soap film.

And you can see that they're not just exact triangles,

you get wonderful curves and arcs

that Otto knew were inherently stable.

Oh, that's lovely, that one there.

The surface tension pulls the strings

into the most sparing shape for each arrangement.

What results is a shape that's not only stable

but remarkably striking too.

So he could make copies of these shapes,

make small little models, which would then be used to construct

the groundbreaking structures you see behind me.

Frei Otto started something of a revolution in architecture.

The sweeping curves of the Munich Stadium

are echoed in countless modern structures.

And although Otto discovered

the mathematical and aesthetic beauty of the Code in the 20th century,

there's evidence that this obsession with form

stretches back thousands of years.

These stone balls were found in Scotland and they date back

to the Neolithic period, which is over 4,000 years ago.

They sit very beautifully in the hands.

They found hundreds of these balls.

But it's not really clear what they were used for.

It's a bit of a mystery.

But imagine the amount of work that's gone into making these shapes.

For example this one here has got four different faces

arranged in a beautifully symmetrical manner.

This one here has six faces, a bit like a cube.

And you can see some of them are really intricate.

This ones got... I don't know how many nodules on there.

Some of them have got up to 160 different nodules.

But these stones really show an obsession with symmetry

and regularity, already, thousands of years ago.

This obsession with shape isn't unique to the ancient Scots.

We find it in other cultures all over the world.

The Egyptians had their pyramids, of course.

But it was the Greeks who first took this innate fascination with shape

and turned it into a subject of its own.

They believed that by understanding its principles,

they could describe the whole world.

And they gave a name to this new idea.

One which meant measuring the Earth.

They called it geometry.

The mainstay of Greek geometry was a discovery of five perfect shapes,

now called the Platonic Solids, after the Greek philosopher Plato,

who believed these were the building blocks of nature.

So we've got the tetrahedron with its four faces,

the cube with its six faces,

the octahedron with its eight faces, the dodecahedron, 12 faces,

and the most complicated shape of all,

the icosahedron, with its 20 faces.

Today these are more commonly known as dice.

We're all used to the familiar six sided dice,

but these four other shapes have also been used as dice for centuries.

What makes them perfect for the job is that they are so regular.

The faces of each are all the same shape. All meet at the same angles.

It means that there's no way of telling one end from another,

and that they are equally likely to land on any face.

But most surprisingly,

these are the only five shapes like this that can possibly exist.

They're the only perfectly symmetrical solids.

It's this almost magical symmetry which made the Greeks believe

that these shapes were so important.

They associated them with the building blocks of nature:

air, fire, earth, the cosmos and water.

These five shapes built the natural world.

It's very easy to dismiss this approach as naive.

After all, it's clear the world around us

isn't made out of just five neat geometric shapes.

But perhaps we should have more faith in this ancient intuition.

Because by laying out the laws of geometry the Greeks had in fact

tapped straight into the Code that shapes all nature.

It turns out that the Greeks were right about their shapes,

but they couldn't have known it, because the world that's governed

by their laws of geometry was completely invisible to them.

We can find traces of it deep underground.

This is the Merkers potash mine,

in the heart of what used to be East Germany.

It has long since stopped production,

but you can still explore its 3,000 miles of tunnels.

That's stunning, my God. I've never seen anything like this.

In fact I think this is the only one like this in the world.

It's absolutely amazing. Just goes on and on down through the cave.

The cave is full of perfectly cubic crystals that mirror

the geometric precision of the Platonic solids.

These cubes are amazing. Look at that.

The surface is perfectly flat

and if you run your finger down the edge here it's so sharp.

Comes down to this precise right angle.

An architect would be happy with that kind of precision.

Doesn't look real.

Even if you look inside you can see

all the cracks are right angles and geometric shapes.

Totally surreal.

Actually, this isn't anything particularly special.

This is just sodium chloride

which we know as salt.

This is what you stick on your chips.

But you don't generally see salt as big a cube as this one here.

How these crystals were able to form with such perfect precision

was a mystery until just over 100 years ago,

when X-rays were discovered.

Our understanding of our biology was transformed

by being able to see inside the human body.

And when X-rays were shone through crystals,

they uncovered another invisible world,

one that was both mysterious and geometric.

This was the world of the atom.

And these neat symmetrical images,

called diffraction patterns,

can reveal how individual atoms were put together

to form the crystals in this cave.

Essentially you've got to think of these a bit like shadows.

Just in the same way as an X-ray of my hand

is a shadow of the bones underneath the skin,

this is a shadow of the billions of atoms contained inside the crystal.

It's a little bit more complicated than that, but essentially,

these are 2D projections of the 3D structure inside this crystal.

So now we can analyse these patterns

and work out exactly how the atoms are arranged inside the salt.

And there is only one possible arrangement of these atoms

that can produce patterns like these.

And it too, unsurprisingly, is a cube.

This is a model of the structure of salt, and these gold balls

are the sodium atoms, and the green ones are the chlorine atoms.

And it's this atomic symmetry which explains

why were seeing such symmetry in these huge crystals.

But instead of just three atoms lining themselves up in this model,

we've got billions and billions of sodium and chlorine atoms

arranging themselves rigidly to create these perfect cubes.

What makes this cave so special

is that the perfect geometric arrangement of the atoms has been

maintained in these huge crystals.

They're a window into nature, and how it's governed by the laws of geometry

at the most fundamental atomic level.

But what's surprising is that we can find the same laws,

not just in rocks and minerals, but deep inside ourselves.

I've come to the Department of Chemical and Structural Biology

at Imperial College in London.

Steve Matthews studies how individual atoms

are built up into living systems, like you and me.

X-rays are obviously very powerful, high energy radiation,

so proteins are very delicate.

So we cool it down with a stream of liquid nitrogen gas

blowing over the crystal.

In this tiny wire loop is another crystal,

but this time, it's a crystal of protein,

part of the machinery of living cells.

Just as it's possible to work out

the atomic structure of the salt crystals with X-rays,

we can deduce the shape of the protein molecules in the same way.

Though the results aren't quite so easy to interpret.

I'd be hard pushed to actually give a name to that shape mathematically.

It looks like a blob.

It doesn't have a shape but many of these blobs

come together to form shapes.

There's a huge amount of structure and symmetry in this protein?

-Oh yes, definitely.

-That's amazing.

We've got a cylinder now.

This is a real surprise to see geometry at work inside our bodies.

But evolution creates a very efficient process,

so symmetry is a very efficient way

of building these types of structures.

So by a process of evolution biology has discovered that...

Before us, yes.

..that geometry gives us the best shapes?

Right. But if you really want symmetry

we can move over to a virus particle.

-I recognise that. That's a icosahedron.

-That's an icosahedron.

This is one of the shapes the Greeks were obsessed with.

-Seems that viruses are too.

-That's right.

It's very striking cos the physical world

you somehow expect maybe salt crystals to be symmetric,

but the biological world everyone considers rather a messy one.

But this is not messy at all. This is beautiful.

The geometric shapes which you find at the heart of our cells

are the most efficient that nature can produce.

It seems like the Greeks could have been right after all.

It's their shapes that build the word around us

and produce its inherent beauty.

'An obsession with symmetry and regulatory.'

The Code dictates some shapes through efficiency...

'The building blocks of nature.'

..and others by providing frameworks for the tiniest particles there are.

'This is nature's code at work.'

'It fits beautifully in the hand.'

What the Greeks discovered in mathematical theory

is to be found at the heart of nature, from crystals to viruses.

It all seems very neat.

'Now I recognise that. That's an icosahedron.'

'The only one like it in the world.'

But our world isn't filled with precise geometric shapes.

It seems random, disordered.

To find out why we need to look to the sky

and the crystals that fall from it.

Snowflakes assemble themselves in the heart of frozen clouds

and fall to earth in a dazzling display.

VOICES CHATTER INAUDIBLY

And if there's one thing we know about snowflakes,

it's that they're all perfectly symmetrical.

-Wow.

-Here we are. It's the snow lab.

Physicist Kenneth Libbrecht has created a lab

for growing and photographing these perfect crystals.

It's a cold chamber. Its actually cold on the bottom, very cold,

about minus 40 on the bottom and about plus 40 on top.

In a sense this machine is trying

to replicate what happens inside a snow cloud.

In a sense, that's right. It's not hard to grow ice crystals.

All you need is cold and water.

In the freezing conditions of the chamber,

we should be able to see the inherent geometry of the world

emerging in front of our eyes, as the crystals start to form.

Now, with any luck, we'll see some stars growing

on the ends of those needles.

As the temperature drops,

billions of water molecules coalesce out of the vapour,

spontaneously arranging themselves into these six pointed patterns.

At least, that's the theory.

But the reality turns out to be very different.

As Ken found out, even in laboratory conditions,

it's almost impossible to grow perfect snowflakes.

I don't think any of these are symmetrical. Not a single one.

What's the chance of getting

a perfectly symmetrical snowflake in here?

PROFESSOR SIGHS

The really beautiful snowflakes are about one in a million.

-Really? Wow.

-Sometimes they've got five sides or three sides.

Five sides? Oh no!

Or three, or sometimes you get a blob.

It's a little hard to see

but this mess here is one funny looking snowflake.

We do tend to think of the snowflake as something

beautifully symmetrical, but actually that's just some

idealised notion and the reality is that they're actually

much more complex and irregular than we think they are.

If the molecular scale it's perfect, but as the crystal gets bigger,

the atoms don't hook on in always exactly the right way,

so when it grows, or how it grows depends on the environment,

the temperature and the humidity, so it starts growing one way,

then moves to a different spot in the cloud and grows a different way

and then a different way, so by the time the crystal hits the ground,

it's had a complex growth history, so it ends up as a complex crystal.

Ah, there it goes.

It seems you can only come so far

in trying to describe the world with simple geometry.

You can see it at work in the salt crystals in the crystal cave.

But in truth, that's one of the very few places in the world

where you'll find such crystals.

The bees use simple geometry to make their honeycomb,

but they've evolved to perform that task over many thousands of years.

And it's only occasionally that you'll ever find a purely symmetrical snowflake.

Because although everything is formed from tidy geometry at the atomic level,

that underlying order falls apart amid all the competing forces of our chaotic world.

Even the Giant's Causeway isn't really a neat hexagonal array.

It's almost there, but amongst the hexagons

there are pentagons, seven-sided columns, even a few with eight sides.

That network of perfectly interlocking hexagons just doesn't exist.

The world clearly isn't just built from simple geometric shapes.

The movement of the sea and the flow of the waves

are far too complicated to explain in these terms.

It's difficult to imagine how we could ever find a code to explain all this complexity.

But what if there are patterns in the chaos of nature?

Patterns that we're not aware of, but that we're attuned to on a subconscious level.

This barn was home to one of the artistic revolutions of the 20th century.

The painter who worked here had become disillusioned with conventional painting techniques.

In fact he stopped painting altogether and started splattering.

He was as controversial as the art he produced.

An arrogant, self-destructive drunk.

And perhaps a visionary.

His name was Jackson Pollock.

The floor you can still see is covered in paint.

What Pollock would do is to lay a canvas out on the floor.

And then - often intoxicated - he would drip and flick the paint all over the surface.

He'd come back week after week, adding more and more layers, more and more colours.

The result was extraordinary.

They're a huge outburst of abstract expressionism.

Just covered in paint, scattered all over the place.

Pollock's paintings sent shockwaves through the art world.

No-one had ever seen anything like this before.

Life Magazine declared him, artist of the century. Others derided his

efforts as the substandard dross of a drunken lunatic.

But though Pollock's paintings courted controversy, they were incredibly influential.

Not least because the apparent random squiggles are strangely compelling.

Many people have tried to copy Pollock's techniques.

Some in homage, others in attempted forgeries.

But nobody seems to be able to reproduce that magic that Pollock brought to the originals.

Pollock's paintings seem to have captured something of the wildness of the natural world.

But for a long time no-one could define exactly what it was that made his work so appealing.

Until it came to the attention of artist and physicist, Richard Taylor.

His unique approach was to invent a machine that can mimic Pollock's eccentric painting style.

It's all based on this apparatus called the Pollockiser.

The Pollockiser? That's lovely.

No, what it is essentially though is what's called a kicked pendulum and as you know a basic pendulum

is very, very regular like a clock, but at the top here what you've got

is a little device that can actually knock the

string as it's swinging around and that induces a very different type of motion called "chaotic motion."

So this would be like Pollock's hand, this would

be what he'd be trying to achieve with that sort of off balance, um,

-painting that we do?

-Absolutely, so they're very similar processes.

-It's very effective.

By recreating his technique, the Pollockiser is able to mimic

one particular aspect of the artist's work.

And that is that it appears more or less the same, no matter how closely you look.

You keep on seeing these patterns unfolding in front of you.

And with a Pollock painting, all of those patterns of different size scales look the same.

This is a property known as fractor.

So if I took pictures at these different scales and showed them to somebody, in some sense they wouldn't

be able to tell which one was the close and which one was far away?

Absolutely. So as long as you can't see that canvas edge, then you have no idea whether you're standing

30 feet away or 2 feet away, they'll both have exactly the same level of complexity.

More than any other painter, Jackson Pollock was able to consistently repeat the same

level of complexity at different scales throughout his paintings.

The fractor quality of his work appeals to us.

Because, despite seeming abstract, it actually mirrors the reality of the world around us.

When we started to actually analyse the buried patterns in there, this amazing thing emerged.

Deep down hidden in there is this level of mathematical structure.

So it's this really delicate interplay between something that looks messy and chaotic, but actually

it has structure and some underlying code hidden inside it?

Absolutely, and you can see it not only in his paintings, but you see it everywhere.

You know like a tree outside.

You look at the tree from far away you see this big trunk with a few branches going off.

Superficially they look cluttered and they look incredibly complex,

but your eye can sense that there's a sort of underlying mathematical structure to all it.

Pollock was the first person to actually

put it on canvas in a direct fashion that no other artist has ever done.

It really is the basic fingerprint of nature.

And that's what's most fascinating about Pollock's art.

In creating work devoid of conventional meaning,

he had in fact stumbled across something fundamental.

Because fractors are how nature builds the world.

Clouds are fractal, because they display the same quality.

Giant clouds are identical to tiny ones.

And it's the same with rocks.

From appearances you can't tell if you're looking at an enormous mountain, or a humble bolder.

And then there are living fractors like this tree.

It's quite easy to see how fractal it is, because if you take one of the branches it looks remarkably like

a small version of the tree itself. If you look at the twigs coming off the branch, they have the same shape.

So you see the same pattern appearing again and again at smaller and smaller scales.

And trees also demonstrate the great powers of fractal systems.

Their great complexity stems from very simple rules.

Now the reason the tree makes this shape is because it wants to maximise the amount of sunlight it gets.

Very clever. But also very simple, because you just need one rule to create this shape.

What the tree does is to grow, then divide. Grow then divide.

And by using this one rule, we get this incredibly complex shape we call a tree.

This is the same pattern repeating itself at a smaller and smaller scale.

It's a rule that's easy to test.

Grow a bit, then branch.

Grow a bit then branch.

And before our eyes a mathematically perfect tree appears.

But just as you never get a perfect snowflake, you never get a perfect tree either.

But allow for some natural variability,

different growing seasons, the wind, an occasional accident and the result is a very real looking tree.

And we find the same fractal branching system time and again throughout nature.

Deep down in there is this level of mathematical structure.

This idea that the patterns

of nature may be inherently fractal was pioneered in the 1970s by French mathematician, Benoit Mandelbrot.

This is his most famous creation.

The Mandelbrot Set.

Its systems of circles and swirls repeats itself at smaller and smaller scales forever.

And this infinite complexity was created from just one very simple mathematical function.

Mandelbrot's quantum leap was to suggest that similar simple mathematical codes

could describe not just trees, but many of the seemingly random shapes of much of the natural world.

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And the most powerful demonstration of that belief comes, not from maths or nature, but from make believe.

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A smart pencil...

In the 1980s, a computer scientist working for the aircraft manufacturer Boeing

was struggling to create computer-generated pictures of planes.

At Boeing, we discovered a method of making curved surfaces,

very nice curved surfaces, so I was applying it to airplanes.

And Boeing publicity photos have mountains behind their planes

and so I wanted to be able to

put a mountain behind my airplane, but I had no idea of the mathematics or how to do that, not a clue.

So you wanted something that however far or near away you were, it would look like something natural?

Yes, exactly, to show that these mountains were

real and live, in the sense that you can move around them with a camera.

So the algorithm needed to be invented

and so that's what I set my mind to doing was invent the algorithm that would produce the mountain pictures.

At the time, even creating a virtual cylinder was hard.

So generating the apparently random jaggedness of a realistic mountain range seemed impossible.

Then Loren found inspiration.

Coincidentally at that time, Mandelbrot's book came out.

He had pictures that showed what fractal mathematics could produce

and so wow, all I have to do is find a way to implement this mathematics

on my computer and I can make pictures of mountains.

Loren set to work to investigate how Mandelbrot's theories about

the real world could be used to make virtual ones.

This is a little film I made in 1980.

-And the landscape is constructed by me, by hand, of about 100 big triangles.

-Yeah.

So that doesn't look very natural.

No, it's very pyramid-like.

So what we're going to do is take each of these big triangles and break it up into little triangles

and break those little triangles up into littler triangles, until

it gets down to the point where you can't see triangles any more.

What Loren had realised was that he could use the maths of fractors

to turn just a handful of triangles into realistic virtual worlds.

We turn the fractal process loose and instantly it looks natural.

We went from about 100 triangles to about 5 million.

And there it is.

And then we jump off the cliff.

You feel that it's a real three-dimensional world.

And we're swooping over the landscape.

Yeah, we're going from ten miles away to ten feet away

and all that detail was generated on the fly as we came in.

-In a few seconds.

-And here's that fractal quality, this infinite complexity at work.

-It's exactly what I wanted.

-Yeah.

By today's standards, this animation does not look like much.

But in the 1980s, no-one had ever seen anything like it.

If you did that by hand, to do that frame by frame, it would take you?

-100 years.

-100 years and this took to generate?

It took about 15 minutes per frame on a computer that's about 100 times slower than my phone.

That one short film changed the face of animation and revolutionised Hollywood.

Loren went on to co-found Pixar,

one of the most successful film studios in the world.

Cars, monsters and, of course, toys owe their existence to the Code.

An empire built on the power of fractors.

Did you realise at the time the potential of the discovery you'd made?

Well, I knew that,

that within a half a second that it was a major discovery.

I've seen, you know, all the special effects, all the movies you can imagine, nothing was like that.

And my heart skipped.

And the power of fractors is still to be hidden in the fabric of Pixar movies.

They use the rule of repetition and self-similarity to create the rocks, clouds and forests.

In fact, the realism and complexity of these virtual worlds is only possible using mathematics.

Fractals are everywhere in these movies.

They generate the texture of the rocks.

And they bring the jungle alive.

That these pretend worlds are so realistic,

demonstrates the power of maths to describe the complexity of nature.

They're evidence that we have glimpsed the Code that governs the shape of the world.

But that Code is a complicated one.

If we want to understand the shape of the world, then we need to recognise

the simple geometry of form at work at the most basic level.

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We need to understand that the universe is lazy.

And that it will always seek out the most efficient solution.

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That at the atomic level, the world is structured around strict geometric laws...

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..that were first recognised by the Greeks thousands of years ago.

We also need to appreciate the complexity of that geometry

playing out against the competing forces of the natural world.

And that means grasping how even the apparent randomness we see around us

is underwritten by mathematical rules like fractors.

Rules that can explain the patterns in everything.

From the chaos of Jackson Pollock's paintings,

to the structure of trees and the realism of virtual worlds.

And that's the beauty of the Code.

However complex we find our world, it provides a reason,

an underlying explanation for why things look and behave as they do.

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This is nature's code of law.

Go to bbc.co.uk/code to find clues

to help you solve the Code's treasure hunt.

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

when you follow the links to the Open University.

Or call:

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