0:00:19 > 0:00:22As a mathematician, I see the world differently.
0:00:22 > 0:00:24POOL BALLS CLICK
0:00:24 > 0:00:29Numbers, shapes, and patterns are everywhere.
0:00:29 > 0:00:31Together, they make up a hidden mathematical world,
0:00:31 > 0:00:35a code that has the power to reveal the inner workings
0:00:35 > 0:00:39of life, the universe, and everything.
0:00:52 > 0:00:56Here in California, beekeeper Steve Godlin is taking me
0:00:56 > 0:01:00to see one of the greatest mysteries in nature.
0:01:06 > 0:01:09That's one of the wonders of the natural world.
0:01:09 > 0:01:10It's beautiful!
0:01:12 > 0:01:15- This almost looks man-made, manufactured.- Yeah.
0:01:15 > 0:01:18It doesn't look like something from the natural world.
0:01:18 > 0:01:23The precision, the fine straight lines they've created.
0:01:23 > 0:01:26The bees' honeycomb is a marvel of natural engineering.
0:01:26 > 0:01:30It's a place to raise their young, and to store their food,
0:01:30 > 0:01:33and it's all made from wax,
0:01:33 > 0:01:35a substance that takes a huge amount of energy
0:01:35 > 0:01:38for them to produce.
0:01:40 > 0:01:42Every cell is identical.
0:01:42 > 0:01:46Six walls, all the same length,
0:01:46 > 0:01:50meeting at exactly 120 degrees.
0:01:50 > 0:01:54The cross section is a perfect six-sided polygon,
0:01:54 > 0:01:56the hexagon.
0:01:57 > 0:02:00With a myriad of natural shapes to choose from,
0:02:00 > 0:02:02why have the bees selected
0:02:02 > 0:02:05this complex geometric structure?
0:02:07 > 0:02:12The answer comes from the bees' need to economise.
0:02:20 > 0:02:24The problem they have to solve is how to create their comb,
0:02:24 > 0:02:26using as little precious wax as possible.
0:02:28 > 0:02:31If they want to produce a network of regular shapes,
0:02:31 > 0:02:34which fit together neatly,
0:02:34 > 0:02:36them you really only have three options.
0:02:36 > 0:02:39You can do equilateral triangles,
0:02:39 > 0:02:41or you could do squares,
0:02:41 > 0:02:43or you can do the bees' hexagons.
0:02:43 > 0:02:48But why, of those three, does the bee choose the hexagons?
0:02:48 > 0:02:53It turns out that the triangles actually use much more wax
0:02:53 > 0:02:55than any of the other shapes.
0:02:55 > 0:02:57Squares are a little better.
0:02:57 > 0:03:00But it's the hexagons that use the least amount of wax.
0:03:00 > 0:03:07It took human mathematicians until 1999 to prove the hexagonal array
0:03:07 > 0:03:10was the most efficient possible solution to this problem.
0:03:12 > 0:03:15Yet, with a little help from evolution,
0:03:15 > 0:03:19the bees worked it out for themselves millions of years ago.
0:03:19 > 0:03:22Perfect hexagons!
0:03:22 > 0:03:25It's easy to see why the hexagon is important to the bees.
0:03:25 > 0:03:28But how on earth do these hard-working creatures,
0:03:28 > 0:03:30with a brain smaller than a sesame seed,
0:03:30 > 0:03:34create the shapes with such precision?
0:03:34 > 0:03:37For that, we need to have a look
0:03:37 > 0:03:40at some of the laziest structures around.
0:03:48 > 0:03:51The soap bubble reveals something fundamental about nature.
0:03:51 > 0:03:54It's lazy. It tries to find the most efficient shape,
0:03:54 > 0:03:56the one using the least energy,
0:03:56 > 0:03:58the least amount of space.
0:04:00 > 0:04:04The sphere is one surface, no corners, infinitely symmetrical.
0:04:10 > 0:04:13When bubbles are on their own, they always try to be spheres,
0:04:13 > 0:04:19but when you start packing them together, their geometry changes.
0:04:23 > 0:04:26The bubbles are incredibly dynamic.
0:04:26 > 0:04:31They're always trying to assume the most efficient shape.
0:04:31 > 0:04:34The one that uses the least energy.
0:04:34 > 0:04:39But if we make each of the bubbles the same size,
0:04:39 > 0:04:41a rather magical shape starts to appear.
0:04:51 > 0:04:52The hexagon.
0:04:53 > 0:04:59This leaves the bees with a very simple way to build their honeycomb.
0:04:59 > 0:05:02All they need to do is build a cylinder of wax,
0:05:02 > 0:05:04and nature will do the rest.
0:05:05 > 0:05:09It's thought that the wax, warmed by their bodies,
0:05:09 > 0:05:12pulls itself into the most efficient configuration.
0:05:14 > 0:05:20When we see that pattern at the heart of the beehive,
0:05:20 > 0:05:24it is, in fact, echoing some of the fundamental laws of the universe.
0:05:26 > 0:05:31This drive to efficiency can be seen written throughout nature.
0:05:35 > 0:05:38It's this hidden geometric force
0:05:38 > 0:05:40that makes the world the shape it is.
0:06:10 > 0:06:13At the heart of the mathematical world, lie numbers.
0:06:13 > 0:06:17They give us the power to describe, measure,
0:06:17 > 0:06:19and count everything in the universe.
0:06:24 > 0:06:26But, numbers aren't always what they seem.
0:06:28 > 0:06:31Take, for example, negative numbers.
0:06:33 > 0:06:37It's impossible to trade anything, stocks, shares, currency,
0:06:37 > 0:06:40even fish, without negative numbers.
0:06:40 > 0:06:42Most of us are comfortable with them.
0:06:42 > 0:06:44Even though we may not like it,
0:06:44 > 0:06:48we understand what it means to have a negative bank balance.
0:06:48 > 0:06:50But, when you start to think about it,
0:06:50 > 0:06:53there's something deeply strange about negative numbers.
0:06:53 > 0:06:58They don't seem to correspond to anything real at all.
0:07:00 > 0:07:03The deeper we look into the mathematical world of numbers,
0:07:03 > 0:07:05the weirder it becomes.
0:07:11 > 0:07:13It's easy to imagine one fish,
0:07:13 > 0:07:15or two fish,
0:07:15 > 0:07:16or no fish at all.
0:07:16 > 0:07:21It's much harder to imagine what minus one fish looks like.
0:07:21 > 0:07:24Negative numbers are so odd that, if I have minus one fish,
0:07:24 > 0:07:25and you give me a fish,
0:07:25 > 0:07:30all you can be certain of is that I've got no fish at all.
0:07:32 > 0:07:36The strange, and powerful thing about numbers,
0:07:36 > 0:07:38is that they can exist in the mathematical world,
0:07:38 > 0:07:42whether or not they seem to make sense in the real world.
0:07:45 > 0:07:47Some numbers are so strange,
0:07:47 > 0:07:51they seem to defy even the laws of mathematics.
0:07:54 > 0:07:58This is one of the most basic facts of mathematics.
0:07:58 > 0:08:01A positive number, multiplied by another positive number
0:08:01 > 0:08:03is a positive number.
0:08:03 > 0:08:05For example, one
0:08:05 > 0:08:07times one
0:08:07 > 0:08:08is one.
0:08:08 > 0:08:12A negative number, multiplied by another negative number,
0:08:12 > 0:08:14also gives a positive number.
0:08:14 > 0:08:20Whenever the signs are the same, the product is always positive.
0:08:20 > 0:08:22However, in the code,
0:08:22 > 0:08:25there's a special number which breaks this rule.
0:08:25 > 0:08:29When I multiply it by itself, it gives the answer minus one.
0:08:29 > 0:08:32This isn't a number I can calculate.
0:08:32 > 0:08:33I can't show you this number.
0:08:33 > 0:08:37Nevertheless, we've given this number a name. It's called "i",
0:08:37 > 0:08:42and it's part of a whole class of numbers called imaginary numbers.
0:08:42 > 0:08:46Introducing these strange new numbers into mathematics
0:08:46 > 0:08:48required an act of the imagination.
0:08:51 > 0:08:54But, despite their strange properties,
0:08:54 > 0:08:57they turn out to have some very practical applications.
0:09:08 > 0:09:11Air traffic control relies on radar to track planes accurately
0:09:11 > 0:09:15during their passage through the air.
0:09:15 > 0:09:19Complex computation is required to decode these signals,
0:09:19 > 0:09:21and distinguish moving objects, like planes,
0:09:21 > 0:09:23from the stationary background.
0:09:23 > 0:09:25CONTROL TOWER CHATTER
0:09:26 > 0:09:30At the heart of that analysis lies i,
0:09:30 > 0:09:32the number that cannot exist.
0:09:34 > 0:09:37You could do these calculations with ordinary numbers.
0:09:37 > 0:09:41But they're so cumbersome, by the time you've done the calculation,
0:09:41 > 0:09:42the plane's moved to somewhere else.
0:09:42 > 0:09:47Altitude 6,000, on a squawk of 7715.
0:09:47 > 0:09:50Using imaginary numbers makes the calculation so much simpler.
0:09:50 > 0:09:53You can track the planes in real time.
0:09:53 > 0:09:54In fact, without them,
0:09:54 > 0:09:57radar would be next to useless for air traffic control.
0:10:03 > 0:10:05Although i is an imaginary number,
0:10:05 > 0:10:08I trust my life to it, every time I get in an aeroplane.
0:10:11 > 0:10:15As strange as it may seem, that's because even the so-called
0:10:15 > 0:10:19imaginary bits of mathematics can be used to explain, control,
0:10:19 > 0:10:22and accurately predict the real world.
0:10:44 > 0:10:47You might think that catching criminals is all about
0:10:47 > 0:10:49finding physical evidence.
0:10:49 > 0:10:51Fingerprints, DNA.
0:10:51 > 0:10:56But, even when these clues aren't available,
0:10:56 > 0:10:59a mathematical fingerprint is left behind.
0:11:02 > 0:11:06In 1888, the most notorious serial killer of all,
0:11:06 > 0:11:10Jack the Ripper, murdered five women in London's East End.
0:11:10 > 0:11:12He was never caught.
0:11:15 > 0:11:18Kim Rossmo has 20 years' experience as a detective inspector,
0:11:18 > 0:11:22and he specialises in catching serial killers.
0:11:25 > 0:11:30Although very rare, these criminals are notoriously hard to track down,
0:11:30 > 0:11:33because they often murder strangers
0:11:33 > 0:11:36in locations they have no obvious connection to.
0:11:36 > 0:11:37It's very common,
0:11:37 > 0:11:39in the investigation of a serial murder case,
0:11:39 > 0:11:43to have hundreds, thousands, even tens of thousands of suspects.
0:11:43 > 0:11:45It's a needle in a haystack problem.
0:11:47 > 0:11:50But Rossmo is no ordinary cop.
0:11:50 > 0:11:53Because he's a brilliant mathematician,
0:11:53 > 0:11:58and uses numbers to understand the patterns criminals leave behind.
0:11:58 > 0:12:01There's logic in how offenders hunt the victim,
0:12:01 > 0:12:02and where they commit the crime.
0:12:02 > 0:12:06If we can decode that, and if we can understand that pattern,
0:12:06 > 0:12:09we can use that information to help us focus a criminal investigation.
0:12:09 > 0:12:13To this day, no-one has been able to identify
0:12:13 > 0:12:17just who Jack the Ripper was, or where he lived.
0:12:17 > 0:12:22But, based purely on the locations the murders took place,
0:12:22 > 0:12:24Rossmo thinks he COULD have tracked him down.
0:12:24 > 0:12:28Because he's noticed patterns in criminal behaviour
0:12:28 > 0:12:31that are as distinctive as a fingerprint.
0:12:31 > 0:12:34And he turned them into a mathematical formula,
0:12:34 > 0:12:36based on probability.
0:12:39 > 0:12:43This formula calculates the likelihood
0:12:43 > 0:12:45that a criminal lives at a specific location,
0:12:45 > 0:12:49based solely on where the crimes took place.
0:12:52 > 0:12:57The first half models what's known as "the least effort principle".
0:12:57 > 0:12:58Inherently, we're all lazy,
0:12:58 > 0:13:01and criminals just as much as anyone else.
0:13:01 > 0:13:04They want to accomplish their goals closer to home,
0:13:04 > 0:13:05rather than further away.
0:13:05 > 0:13:08The probability gradually decreases
0:13:08 > 0:13:10the further you get from the crimes.
0:13:13 > 0:13:15The second half of the equation
0:13:15 > 0:13:18describes something called "the buffer zone".
0:13:18 > 0:13:22Criminals avoid committing crimes too close to home,
0:13:22 > 0:13:24for fear of drawing attention to themselves.
0:13:27 > 0:13:30It's the interaction of these two behaviours
0:13:30 > 0:13:33that allows Rossmo to calculate the most probable location
0:13:33 > 0:13:35of the criminal's home.
0:13:36 > 0:13:40So, naturally, I was very interested in what would happen
0:13:40 > 0:13:43if I entered the locations of the five linked crimes
0:13:43 > 0:13:46into the equation.
0:13:47 > 0:13:51With a computer programme based on his formula, Rossmo is creating
0:13:51 > 0:13:55a geographic profile to show the hotspots where the Ripper
0:13:55 > 0:13:57was most likely to have lived.
0:14:04 > 0:14:08It was this geographic profile that led us here,
0:14:08 > 0:14:10to Flower & Dean Street.
0:14:10 > 0:14:15Flower & Dean Street should have been the epicentre of their search.
0:14:16 > 0:14:20Do you think if you'd been alive at the end of the 19th century,
0:14:20 > 0:14:24and you'd had this equation, this guy might have been found?
0:14:24 > 0:14:26Knowing what we know today of serial killers,
0:14:26 > 0:14:29and with modern forensic techniques such as DNA,
0:14:29 > 0:14:32I'm pretty sure Jack the Ripper WOULD have been caught,
0:14:32 > 0:14:35if he was committing his crimes today.
0:14:35 > 0:14:39We may never know the truth about Jack the Ripper,
0:14:39 > 0:14:41but Rossmo's technique of geographic profiling
0:14:41 > 0:14:45has been used time and time again to help police all over the world
0:14:45 > 0:14:50narrow their search, from an entire city, to just a handful of streets.
0:14:54 > 0:14:57And at its heart lies pure mathematics.
0:15:17 > 0:15:21With the world's fish stocks under pressure, it's vital for us
0:15:21 > 0:15:24to find out as much about their populations as we possibly can.
0:15:25 > 0:15:29But with so many fish out there, it seems an impossible task.
0:15:30 > 0:15:33Yet using mathematics, I can discover things
0:15:33 > 0:15:38about the inhabitants of our oceans without even getting my feet wet.
0:15:44 > 0:15:47I started fishing in Brighton in 1972.
0:15:47 > 0:15:51Been a fisherman for 40 years, catching Dover sole.
0:15:53 > 0:15:57That's the main target species for the English Channel.
0:15:57 > 0:15:59Each time he goes out fishing,
0:15:59 > 0:16:04Sam Brenchley catches Dover sole of all different shapes and sizes.
0:16:06 > 0:16:09And by tapping into the power of mathematics, I can predict
0:16:09 > 0:16:14a weight for the largest fish Sam has caught in his entire career.
0:16:14 > 0:16:18All I need to do is get my hands on today's catch.
0:16:26 > 0:16:28That's 180 grams.
0:16:28 > 0:16:31'Even though I've only got a handful of fish,
0:16:31 > 0:16:34'their weights will give me all the information
0:16:34 > 0:16:37'I need for my prediction.'
0:16:37 > 0:16:41She's back! Now using these numbers, I can calculate
0:16:41 > 0:16:48that the largest one should be about 1.3 kg, which is roughly 3 lbs.
0:16:48 > 0:16:53'I've not weighed a single fish anywhere near that size
0:16:53 > 0:16:55'so let's see if I'm right.'
0:16:55 > 0:16:59What's the largest Dover sole you've caught in your career?
0:16:59 > 0:17:02We call them doormats, the large ones.
0:17:02 > 0:17:05And...you maybe get four or five a season.
0:17:05 > 0:17:09The largest I would say is 3 to 3.5 lbs.
0:17:09 > 0:17:14It's always nice to catch big stuff, you know? Well, I think it is anyway.
0:17:20 > 0:17:24So without ever getting my hands on one of these giant doormats,
0:17:24 > 0:17:26just how did I work it out?
0:17:27 > 0:17:32Now, the reason this compilation is possible is because the distribution
0:17:32 > 0:17:35of the weights of fish, in fact the distribution of lots of things
0:17:35 > 0:17:41like the height of people in the UK or IQ, is given by this formula.
0:17:42 > 0:17:46This is the normal distribution equation
0:17:46 > 0:17:49and it's one of the most important bits of mathematics
0:17:49 > 0:17:52for understanding variation in the natural world.
0:17:52 > 0:17:56And it describes the shape of a graph that pops up time
0:17:56 > 0:18:00and time again throughout nature, the bell curve.
0:18:01 > 0:18:05The area under the curve represents all the fish Sam's ever caught.
0:18:07 > 0:18:10Most of them will be an average size.
0:18:10 > 0:18:14The small tiddlers and large doormats are much less likely.
0:18:16 > 0:18:20To put values to this graph, I just need two bits of information,
0:18:20 > 0:18:24the mean shows me where the centre of the bell curve lies
0:18:24 > 0:18:28and a standard deviation shows me the range of the weights.
0:18:28 > 0:18:33I approximated both of these just by weighing Sam's catch.
0:18:33 > 0:18:36Together, they allowed me to estimate values for all
0:18:36 > 0:18:40the fish he's ever caught, from the smallest to the largest.
0:18:44 > 0:18:48Knowing the weight of Sam's largest fish might not seem important.
0:18:48 > 0:18:52But this technique allows us to discover things about all manner
0:18:52 > 0:18:56of populations by measuring just a small sample.
0:18:58 > 0:19:01And from biology to medicine and even engineering,
0:19:01 > 0:19:03the bell curve gives us
0:19:03 > 0:19:07the power to make predictions about our world and everything in it.
0:19:24 > 0:19:28Right now, across the planet, people are unconsciously building
0:19:28 > 0:19:31one of the largest databases in the world.
0:19:31 > 0:19:33Every time we go online,
0:19:33 > 0:19:37whether it's to check an e-mail or find something out,
0:19:37 > 0:19:42we're all adding to an already overwhelming mass of information.
0:19:42 > 0:19:45A random jumble of data about our inner thoughts, interests
0:19:45 > 0:19:47and future plans.
0:19:50 > 0:19:54Yet when we start to look closely at all this complexity,
0:19:54 > 0:19:57surprising patterns begin to emerge...
0:19:58 > 0:20:01..patterns that can reveal things about our future
0:20:01 > 0:20:03and even save lives.
0:20:06 > 0:20:08With over 91 million web searches a day,
0:20:08 > 0:20:12Google have access to a constantly updating stream of information.
0:20:14 > 0:20:18Our random search entries might seem useless at first
0:20:18 > 0:20:21but they found ways to tap into this data.
0:20:21 > 0:20:25Think of all the things people might search for on a daily basis.
0:20:25 > 0:20:29Think of the things that YOU might search for on a daily basis.
0:20:29 > 0:20:32You, I've searched for a couple of cities in Mexico
0:20:32 > 0:20:34and films in Hackney today.
0:20:34 > 0:20:38Lots of people may be searching for a similar thing,
0:20:38 > 0:20:40movies and Hackney for example.
0:20:40 > 0:20:43And you can see if you look at that query over the past three years,
0:20:43 > 0:20:48what would the pattern of searches for that term look like?
0:20:49 > 0:20:51Their researchers had a hunch
0:20:51 > 0:20:53that if they could decode people's searches,
0:20:53 > 0:20:58they'd be able to make predictions about the real world.
0:20:58 > 0:21:00And they decided to test their theory
0:21:00 > 0:21:03with the global killer influenza,
0:21:03 > 0:21:06a virus that's responsible for hundreds of thousands
0:21:06 > 0:21:08of deaths every year.
0:21:10 > 0:21:12So, flu has a nice, seasonal pattern
0:21:12 > 0:21:17and because it has that pattern every year, over many years,
0:21:17 > 0:21:21we're able to take that trend and say
0:21:21 > 0:21:25which search queries match that pattern.
0:21:25 > 0:21:28Although flu tends to come back year after year,
0:21:28 > 0:21:32it's very hard to predict exactly when and where it's going to strike.
0:21:32 > 0:21:35If we knew, hospitals, doctors
0:21:35 > 0:21:40and health organisations across the world could better prepare.
0:21:41 > 0:21:45So the team analysed the pattern of every search term
0:21:45 > 0:21:47in their database from the last five years,
0:21:47 > 0:21:51looking for those that appeared more frequently during flu outbreaks.
0:21:51 > 0:21:54And unsurprisingly, when people had flu,
0:21:54 > 0:21:57there was an increase in flu-related searches.
0:21:57 > 0:22:01So people were searching for things like symptoms
0:22:01 > 0:22:04or medications or sore throat.
0:22:04 > 0:22:07There are other things like complications.
0:22:10 > 0:22:15But what no-one expected was that this correlation would be so exact.
0:22:15 > 0:22:19Flu-related terms rose and fell perfectly in line
0:22:19 > 0:22:21with real-life flu outbreaks.
0:22:21 > 0:22:24There's an accurate data of flu activity based
0:22:24 > 0:22:27on lots of people searching for these terms.
0:22:27 > 0:22:30We were amazed by this finding.
0:22:30 > 0:22:34This allowed them to build a model based solely on search terms
0:22:34 > 0:22:37that could reliably detect and predict
0:22:37 > 0:22:39the presence of flu epidemics in real time.
0:22:43 > 0:22:46Google flu trends can spot an outbreak of flu before people
0:22:46 > 0:22:50have even gone to the doctor.
0:22:50 > 0:22:54Just 20 years ago, these kinds of predictions were unthinkable.
0:22:54 > 0:22:56The data just didn't exist.
0:22:58 > 0:23:01But flu trends proves that the vast quantity of seemingly random data
0:23:01 > 0:23:06created by our increasingly connected lives isn't useless.
0:23:06 > 0:23:09Using mathematics, we can make sense of it
0:23:09 > 0:23:12and create tools with the power to save lives.
0:23:34 > 0:23:38Mathematics reveals itself in some spectacular ways,
0:23:38 > 0:23:41from the bees' honeycomb
0:23:41 > 0:23:43and the hexagonal snowflake
0:23:43 > 0:23:46to these remarkable salt crystals.
0:23:49 > 0:23:52But these are the rare exceptions.
0:23:52 > 0:23:57Most of the natural world is complex and seems random.
0:23:57 > 0:24:01It's hard to believe that we could use mathematics
0:24:01 > 0:24:03to explain all this apparent chaos.
0:24:08 > 0:24:12But even things that look disordered like these trees
0:24:12 > 0:24:14do have a hidden mathematical order.
0:24:17 > 0:24:20Now the reason the tree makes this shape
0:24:20 > 0:24:23is because it wants to maximise the amount of sunlight it gets.
0:24:23 > 0:24:26Very clever but also very simple
0:24:26 > 0:24:28because you just need one rule to create this shape.
0:24:30 > 0:24:34And it's easy to demonstrate this rule using computer graphics.
0:24:36 > 0:24:40Grow a bit, then branch. Grow a bit, then branch.
0:24:41 > 0:24:46Repeat this one rule time and time again and before our eyes,
0:24:46 > 0:24:49a mathematically-perfect tree appears.
0:24:52 > 0:24:56And allow for some natural variations, different seasons,
0:24:56 > 0:24:58the wind and occasional accidents,
0:24:58 > 0:25:02and the result is a very real looking tree.
0:25:06 > 0:25:09You can imagine this rule repeating itself for ever,
0:25:09 > 0:25:11branching into infinity.
0:25:16 > 0:25:20In the 1970s, French mathematician Benoit Mandelbrot
0:25:20 > 0:25:23set about studying these hypnotic computer-generated patterns.
0:25:29 > 0:25:34Just like a tree, they were created from one very simple rule,
0:25:34 > 0:25:36set up to repeat itself time and time again.
0:25:39 > 0:25:43But these shapes go on for ever, creating infinite complexity.
0:25:44 > 0:25:47It's a property known as fractal.
0:25:49 > 0:25:52Mandelbrot believed that fractals could be used to describe
0:25:52 > 0:25:56many of the seemingly random shapes we see in the natural world.
0:26:00 > 0:26:03And the most powerful demonstration of that belief
0:26:03 > 0:26:08comes not from mathematics or nature but from the world of make-believe.
0:26:11 > 0:26:16Over 30 years ago, Loren Carpenter made it two minute film
0:26:16 > 0:26:19that would revolutionise the world of animation.
0:26:20 > 0:26:23This is a little film I made in 1980.
0:26:23 > 0:26:28The landscape is constructed by me, by hand, of about 100 big triangles.
0:26:28 > 0:26:31- That doesn't look very natural. - No, it's very pyramid-like.
0:26:31 > 0:26:36What we're going to do is take each of these big triangles and break it up into little triangles
0:26:36 > 0:26:40and break those little triangles up into littler triangles.
0:26:40 > 0:26:43Until that gets down to the point where you can't see triangles any more.
0:26:43 > 0:26:46What Loren had realised was that he could use
0:26:46 > 0:26:49the mathematics of fractals to turn just a handful of triangles
0:26:49 > 0:26:52into realistic virtual worlds.
0:26:54 > 0:26:58We went from about 100 triangles to about five million.
0:27:00 > 0:27:05We turned the fractal process loose and instantly it looks natural.
0:27:05 > 0:27:10And here's that fractal quality, this infinite complexity at work.
0:27:10 > 0:27:13That's exactly what I wanted, yeah.
0:27:15 > 0:27:18By today's standards, this animation doesn't look like much
0:27:18 > 0:27:24but in the 1980s, no-one had seen anything like it before.
0:27:24 > 0:27:28Loren Carpenter had used fractals to revolutionise computer graphics.
0:27:33 > 0:27:36It was because of this one short film that Loren went on to co-found
0:27:36 > 0:27:42one of the most successful film studios in the world, Pixar.
0:27:45 > 0:27:48This empire of cars and monsters
0:27:48 > 0:27:51and toys, was built on the power of fractals.
0:27:57 > 0:28:01And fractals are still used at Pixar today.
0:28:01 > 0:28:05They create the surface of rocks, the texture of clouds
0:28:05 > 0:28:08and bring the trees and forests alive.
0:28:11 > 0:28:15The fact that these virtual worlds are so realistic demonstrates
0:28:15 > 0:28:20the power of mathematics to describe the infinite complexity of nature.
0:28:28 > 0:28:30Subtitles by Red Bee Media Ltd
0:28:30 > 0:28:33E-mail subtitling@bbc.co.uk