The Code

As a mathematician, I see the world differently.

POOL BALLS CLICK

Numbers, shapes, and patterns are everywhere.

Together, they make up a hidden mathematical world,

a code that has the power to reveal the inner workings

of life, the universe, and everything.

Here in California, beekeeper Steve Godlin is taking me

to see one of the greatest mysteries in nature.

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

It's beautiful!

-This almost looks man-made, manufactured.

-Yeah.

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

The precision, the fine straight lines they've created.

The bees' honeycomb is a marvel of natural engineering.

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

and it's all made from wax,

a substance that takes a huge amount of energy

for them to produce.

Every cell is identical.

Six walls, all the same length,

meeting at exactly 120 degrees.

The cross section is a perfect six-sided polygon,

the hexagon.

With a myriad of natural shapes to choose from,

why have the bees selected

this complex geometric structure?

The answer comes from the bees' need to economise.

The problem they have to solve is how to create their comb,

using as little precious wax as possible.

If they want to produce a network of regular shapes,

which fit together neatly,

them you really only have 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?

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 that use the least amount of wax.

It took human mathematicians until 1999 to prove the hexagonal array

was the most efficient possible solution to this problem.

Yet, with a little help from evolution,

the bees worked it out for themselves millions of years ago.

Perfect hexagons!

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

But how on earth do these hard-working creatures,

with a brain smaller than a sesame seed,

create the shapes with such precision?

For that, we need to have a look

at some of the laziest structures around.

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.

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

When bubbles are on their own, they always try to be spheres,

but when you start packing them together, their geometry changes.

The bubbles are incredibly dynamic.

They're always trying to assume the most efficient shape.

The one that uses the least energy.

But if we make each of the bubbles the same size,

a rather magical shape starts to appear.

The hexagon.

This leaves the bees with a very simple way to build their honeycomb.

All they need to do is build a cylinder of wax,

and nature will do the rest.

It's thought that the wax, warmed by their bodies,

pulls itself into the most efficient configuration.

When we see that pattern at the heart of the beehive,

it is, in fact, echoing some of the fundamental laws of the universe.

This drive to efficiency can be seen written throughout nature.

It's this hidden geometric force

that makes the world the shape it is.

At the heart of the mathematical world, lie numbers.

They give us the power to describe, measure,

and count everything in the universe.

But, numbers aren't always what they seem.

Take, for example, negative numbers.

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

even fish, without negative numbers.

Most of us are comfortable with them.

Even though we may not like it,

we understand what it means to have a negative bank balance.

But, when you start to think about it,

there's something deeply strange about negative numbers.

They don't seem to correspond to anything real at all.

The deeper we look into the mathematical world of numbers,

the weirder it becomes.

It's easy to imagine one fish,

or two fish,

or no fish at all.

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

Negative numbers are so odd that, if I have minus one fish,

and you give me a fish,

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

The strange, and powerful thing about numbers,

is that they can exist in the mathematical world,

whether or not they seem to make sense in the real world.

Some numbers are so strange,

they seem to defy even the laws of mathematics.

This is one of the most basic facts of mathematics.

A positive number, multiplied by another positive number

is a positive number.

For example, one

times one

is one.

A negative number, multiplied by another negative number,

also gives a positive number.

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

However, in the code,

there's a special number which breaks this rule.

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

This isn't a number I can calculate.

I can't show you this number.

Nevertheless, we've given this number a name. It's called "i",

and it's part of a whole class of numbers called imaginary numbers.

Introducing these strange new numbers into mathematics

required an act of the imagination.

But, despite their strange properties,

they turn out to have some very practical applications.

Air traffic control relies on radar to track planes accurately

during their passage through the air.

Complex computation is required to decode these signals,

and distinguish moving objects, like planes,

from the stationary background.

CONTROL TOWER CHATTER

At the heart of that analysis lies i,

the number that cannot exist.

You could do these calculations with ordinary numbers.

But they're so cumbersome, by the time you've done the calculation,

the plane's moved to somewhere else.

Altitude 6,000, on a squawk of 7715.

Using imaginary numbers makes the calculation so much simpler.

You can track the planes in real time.

In fact, without them,

radar would be next to useless for air traffic control.

Although i is an imaginary number,

I trust my life to it, every time I get in an aeroplane.

As strange as it may seem, that's because even the so-called

imaginary bits of mathematics can be used to explain, control,

and accurately predict the real world.

You might think that catching criminals is all about

finding physical evidence.

Fingerprints, DNA.

But, even when these clues aren't available,

a mathematical fingerprint is left behind.

In 1888, the most notorious serial killer of all,

Jack the Ripper, murdered five women in London's East End.

He was never caught.

Kim Rossmo has 20 years' experience as a detective inspector,

and he specialises in catching serial killers.

Although very rare, these criminals are notoriously hard to track down,

because they often murder strangers

in locations they have no obvious connection to.

It's very common,

in the investigation of a serial murder case,

to have hundreds, thousands, even tens of thousands of suspects.

It's a needle in a haystack problem.

But Rossmo is no ordinary cop.

Because he's a brilliant mathematician,

and uses numbers to understand the patterns criminals leave behind.

There's logic in how offenders hunt the victim,

and where they commit the crime.

If we can decode that, and if we can understand that pattern,

we can use that information to help us focus a criminal investigation.

To this day, no-one has been able to identify

just who Jack the Ripper was, or where he lived.

But, based purely on the locations the murders took place,

Rossmo thinks he COULD have tracked him down.

Because he's noticed patterns in criminal behaviour

that are as distinctive as a fingerprint.

And he turned them into a mathematical formula,

based on probability.

This formula calculates the likelihood

that a criminal lives at a specific location,

based solely on where the crimes took place.

The first half models what's known as "the least effort principle".

Inherently, we're all lazy,

and criminals just as much as anyone else.

They want to accomplish their goals closer to home,

rather than further away.

The probability gradually decreases

the further you get from the crimes.

The second half of the equation

describes something called "the buffer zone".

Criminals avoid committing crimes too close to home,

for fear of drawing attention to themselves.

It's the interaction of these two behaviours

that allows Rossmo to calculate the most probable location

of the criminal's home.

So, naturally, I was very interested in what would happen

if I entered the locations of the five linked crimes

into the equation.

With a computer programme based on his formula, Rossmo is creating

a geographic profile to show the hotspots where the Ripper

was most likely to have lived.

It was this geographic profile that led us here,

to Flower & Dean Street.

Flower & Dean Street should have been the epicentre of their search.

Do you think if you'd been alive at the end of the 19th century,

and you'd had this equation, this guy might have been found?

Knowing what we know today of serial killers,

and with modern forensic techniques such as DNA,

I'm pretty sure Jack the Ripper WOULD have been caught,

if he was committing his crimes today.

We may never know the truth about Jack the Ripper,

but Rossmo's technique of geographic profiling

has been used time and time again to help police all over the world

narrow their search, from an entire city, to just a handful of streets.

And at its heart lies pure mathematics.

With the world's fish stocks under pressure, it's vital for us

to find out as much about their populations as we possibly can.

But with so many fish out there, it seems an impossible task.

Yet using mathematics, I can discover things

about the inhabitants of our oceans without even getting my feet wet.

I started fishing in Brighton in 1972.

Been a fisherman for 40 years, catching Dover sole.

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

Each time he goes out fishing,

Sam Brenchley catches Dover sole of all different shapes and sizes.

And by tapping into the power of mathematics, I can predict

a weight for the largest fish Sam has caught in his entire career.

All I need to do is get my hands on today's catch.

That's 180 grams.

'Even though I've only got a handful of fish,

'their weights will give me all the information

'I need for my prediction.'

She's back! Now using these numbers, I can calculate

that the largest one should be about 1.3 kg, which is roughly 3 lbs.

'I've not weighed a single fish anywhere near that size

'so let's see if I'm right.'

What's the largest Dover sole you've caught in your career?

We call them doormats, the large ones.

And...you maybe get four or five a season.

The largest I would say is 3 to 3.5 lbs.

It's always nice to catch big stuff, you know? Well, I think it is anyway.

So without ever getting my hands on one of these giant doormats,

just how did I work it out?

Now, the reason this compilation is possible is because the distribution

of the weights of fish, in fact the distribution of lots of things

like the height of people in the UK or IQ, is given by this formula.

This is the normal distribution equation

and it's one of the most important bits of mathematics

for understanding variation in the natural world.

And it describes the shape of a graph that pops up time

and time again throughout nature, the bell curve.

The area under the curve represents all the fish Sam's ever caught.

Most of them will be an average size.

The small tiddlers and large doormats are much less likely.

To put values to this graph, I just need two bits of information,

the mean shows me where the centre of the bell curve lies

and a standard deviation shows me the range of the weights.

I approximated both of these just by weighing Sam's catch.

Together, they allowed me to estimate values for all

the fish he's ever caught, from the smallest to the largest.

Knowing the weight of Sam's largest fish might not seem important.

But this technique allows us to discover things about all manner

of populations by measuring just a small sample.

And from biology to medicine and even engineering,

the bell curve gives us

the power to make predictions about our world and everything in it.

Right now, across the planet, people are unconsciously building

one of the largest databases in the world.

Every time we go online,

whether it's to check an e-mail or find something out,

we're all adding to an already overwhelming mass of information.

A random jumble of data about our inner thoughts, interests

and future plans.

Yet when we start to look closely at all this complexity,

surprising patterns begin to emerge...

..patterns that can reveal things about our future

and even save lives.

With over 91 million web searches a day,

Google have access to a constantly updating stream of information.

Our random search entries might seem useless at first

but they found ways to tap into this data.

Think of all the things people might search for on a daily basis.

Think of the things that YOU might search for on a daily basis.

You, I've searched for a couple of cities in Mexico

and films in Hackney today.

Lots of people may be searching for a similar thing,

movies and Hackney for example.

And you can see if you look at that query over the past three years,

what would the pattern of searches for that term look like?

Their researchers had a hunch

that if they could decode people's searches,

they'd be able to make predictions about the real world.

And they decided to test their theory

with the global killer influenza,

a virus that's responsible for hundreds of thousands

of deaths every year.

So, flu has a nice, seasonal pattern

and because it has that pattern every year, over many years,

we're able to take that trend and say

which search queries match that pattern.

Although flu tends to come back year after year,

it's very hard to predict exactly when and where it's going to strike.

If we knew, hospitals, doctors

and health organisations across the world could better prepare.

So the team analysed the pattern of every search term

in their database from the last five years,

looking for those that appeared more frequently during flu outbreaks.

And unsurprisingly, when people had flu,

there was an increase in flu-related searches.

So people were searching for things like symptoms

or medications or sore throat.

There are other things like complications.

But what no-one expected was that this correlation would be so exact.

Flu-related terms rose and fell perfectly in line

with real-life flu outbreaks.

There's an accurate data of flu activity based

on lots of people searching for these terms.

We were amazed by this finding.

This allowed them to build a model based solely on search terms

that could reliably detect and predict

the presence of flu epidemics in real time.

Google flu trends can spot an outbreak of flu before people

have even gone to the doctor.

Just 20 years ago, these kinds of predictions were unthinkable.

The data just didn't exist.

But flu trends proves that the vast quantity of seemingly random data

created by our increasingly connected lives isn't useless.

Using mathematics, we can make sense of it

and create tools with the power to save lives.

Mathematics reveals itself in some spectacular ways,

from the bees' honeycomb

and the hexagonal snowflake

to these remarkable salt crystals.

But these are the rare exceptions.

Most of the natural world is complex and seems random.

It's hard to believe that we could use mathematics

to explain all this apparent chaos.

But even things that look disordered like these trees

do have a hidden mathematical order.

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.

And it's easy to demonstrate this rule using computer graphics.

Grow a bit, then branch. Grow a bit, then branch.

Repeat this one rule time and time again and before our eyes,

a mathematically-perfect tree appears.

And allow for some natural variations, different seasons,

the wind and occasional accidents,

and the result is a very real looking tree.

You can imagine this rule repeating itself for ever,

branching into infinity.

In the 1970s, French mathematician Benoit Mandelbrot

set about studying these hypnotic computer-generated patterns.

Just like a tree, they were created from one very simple rule,

set up to repeat itself time and time again.

But these shapes go on for ever, creating infinite complexity.

It's a property known as fractal.

Mandelbrot believed that fractals could be used to describe

many of the seemingly random shapes we see in the natural world.

And the most powerful demonstration of that belief

comes not from mathematics or nature but from the world of make-believe.

Over 30 years ago, Loren Carpenter made it two minute film

that would revolutionise the world of animation.

This is a little film I made in 1980.

The landscape is constructed by me, by hand, of about 100 big triangles.

-That doesn't look very natural.

-No, it's very pyramid-like.

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

What Loren had realised was that he could use

the mathematics of fractals to turn just a handful of triangles

into realistic virtual worlds.

We went from about 100 triangles to about five million.

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

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

That's exactly what I wanted, yeah.

By today's standards, this animation doesn't look like much

but in the 1980s, no-one had seen anything like it before.

Loren Carpenter had used fractals to revolutionise computer graphics.

It was because of this one short film that Loren went on to co-found

one of the most successful film studios in the world, Pixar.

This empire of cars and monsters

and toys, was built on the power of fractals.

And fractals are still used at Pixar today.

They create the surface of rocks, the texture of clouds

and bring the trees and forests alive.

The fact that these virtual worlds are so realistic demonstrates

the power of mathematics to describe the infinite complexity of nature.

Subtitles by Red Bee Media Ltd

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