Original subtitles

NARRATOR: You can find it in a rain forest,

on the frontiers of medical research,

in the movies,

and it's all over the world of wireless communications.

One of nature's biggest design secrets

has finally been revealed.

My God! Of course, it's obvious.

NARRATOR: It's an odd-looking shape you may never have heard of,

but it's everywhere around you,

the jagged, repeating form called a fractal.

They're all over in biology.

They are solutions that natural selection has come up with

over and over and over again.

NARRATOR: Fractals are in our lungs,

kidneys, and blood vessels.

Flowers, plants,

weather systems,

the rhythms of the heart,

the very essences of life.

NARRATOR: But it took a maverick mathematician

to figure out how they work.

I don't play with formulas; I play with pictures,

and that is what I've been doing all my life.

NARRATOR: His was a bold challenge to centuries-old assumptions

about the various forms that nature takes.

The blinders came off and people could see forms

that were always there, but formerly were invisible.

NARRATOR: Making the invisible visible.

Finding order in disorder.

What mysteries can it help us unravel?

Coming up next on NOVA:

"Hunting the Hidden Dimension."

Captioning sponsored by EXXON MOBIL,

DAVID H. KOCH,

the HOWARD HUGHES MEDICAL INSTITUTE,

the CORPORATION FOR PUBLIC BROADCASTING

and VIEWERS LIKE YOU.

Major funding for NOVA is provided by the following:

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NARRATOR: In 1978, at Boeing Aircraft in Seattle,

engineers were designing experimental aircraft.

Exotic things with two wings or two tails

or two fuselages, just weird stuff.

'Cause who knows, it might work.

NARRATOR: A young computer scientist named Loren Carpenter

was helping them visualize

what the planes might look like in flight.

CARPENTER: I would get the data from them

and make pictures, uh, from various angles.

But I wanted to be able to put a mountain behind them,

because every Boeing publicity photo in existence

has a mountain behind it.

But there was no way to do mountains.

Mountains had millions and millions of little triangles

or polygons, or whatever you want to call it,

and, uh, we had enough trouble with a hundred.

Especially in those days when our machines were, uh,

slower than the ones you have in your watch.

NARRATOR: Carpenter didn't want to make just any mountains.

He wanted to create a landscape the planes could fly through;

but there was no way to do that

with existing animation techniques.

From the time movies began,

animators had to draw each frame by hand--

thousands of them to make even a short cartoon.

(echoing): That's why they call me Thumper.

NARRATOR: But that was before Loren Carpenter stumbled across

the work of a little-known mathematician

named Benoit Mandelbrot.

CARPENTER: In 1978, I ran into this book at a bookstore,

Fractals: Form, Chance, and Dimension

by Benoit Mandelbrot,

and it has to do with the fractal geometry of nature.

So I bought the book, took it home and read it--

cover to cover, every last little word,

including the footnotes and references-- twice.

NARRATOR: In his book Mandelbrot said that many forms in nature

can be described mathematically as "fractals,"

a word he invented to define shapes

that look jagged and broken.

He said that you can create a fractal

by taking a smooth-looking shape and breaking it into pieces,

over and over again.

Carpenter decided he'd try doing that on his computer.

CARPENTER: Within three days,

I was producing pictures of mountains

on my computer at work.

The method is dead-simple.

You start with a landscape made out of very rough triangles,

big ones, and then for each triangle,

break it into four triangles, and then do that again,

and again and again...

NARRATOR: Endless repetition--

what mathematicians call "iteration."

It's one of the keys to fractal geometry.

CARPENTER: The pictures were stunning.

They were just totally stunning.

No one has ever seen anything like this,

and I just opened a whole new door

to a new world of making pictures.

And it got the computer graphics community excited about fractals

because suddenly, they were easy to do.

And so people started doing them all over the place.

NARRATOR: Carpenter soon left Boeing to join Lucasfilm,

where, instead of making mountains,

he created a whole new planet

for Star Trek II: The Wrath of Khan.

It was the first-ever completely computer-generated sequence

in a feature film...

Fascinating.

NARRATOR: ...made possible by the new mathematics of fractal geometry.

Benoit Mandelbrot, whose work had inspired that innovation,

was someone who prided himself

on standing outside the mainstream.

I can see things that nobody else suspects

until I point out to them.

"Oh, of course, of course."

But they haven't seen it before.

NARRATOR: You can see it in the clouds,

in the mountains,

even inside the human body.

The key to fractal geometry and the thing that evaded anyone

until, really, Mandelbrot sort of said

this is the way to look at things, is that...

if you look on the surface,

you see complexity, and it looks very non-mathematical.

What Mandelbrot said was that think not of what you see

but what it took to produce what you see.

NARRATOR: It takes endless repetition

and that gives rise to one of the defining characteristics

of a fractal,

what mathematicians call "self-similarity."

The main idea is always, as you zoom in and zoom out,

the objects look the same.

If you look at something at this scale...

and then you pick a small piece of it and you zoom in,

it looks very much the same.

NARRATOR: The whole of the fractal looks just like a part,

which looks just like the next smaller part.

The similarity of the pattern just keeps on going.

One of the most familiar examples

of self-similarity is a tree.

If we look at each of the nodes,

the branching nodes of this tree,

what you'll actually see is that the pattern of branching

is very similar throughout the tree.

As we go from the base of the tree to higher up,

you'll see we'll have mother branches

and branching then into daughter branches.

If we take this one branch and node

and then go up to a higher branch or node,

what we'll actually find is, again,

that the pattern of branching is similar.

Again, this pattern of branching is repeated throughout the tree,

all the way, ultimately, out to the tips,

where the leaves are.

NARRATOR: You see self-similarity in everything:

from a stalk of broccoli

to the surface of the moon,

to the arteries that transport blood through our bodies.

But Mandelbrot's fascination

with these irregular-looking shapes

put him squarely at odds

with centuries of mathematical tradition.

In the whole of science, the whole of mathematics,

a smoothness with everything.

What I did was to open up roughness for investigation.

DEVLIN: We used mathematics to build the pyramids,

to construct the Parthenon.

We use mathematics to study

the regular motion of the planets

and so forth.

We became used to the fact that certain patterns

were amenable to mathematics-- the architectural ones,

largely the patterns of human-made structures,

where we had straight lines and circles

and the perfect geometric shapes.

The basic assumption that underlies classical mathematics

is that everything is extremely regular.

I mean, you reduce everything to straight lines.

Circles, triangles.

Flat surfaces.

Pyramids,

tetrahedrons, icosahedrons, dodecahedrons.

Smooth edges.

DEVLIN: Classical mathematics is really only well-suited to study

the world that we've created,

the things we've built using that classical mathematics.

The patterns in nature,

the things that were already there

before we came onto the planet,

the trees, the plants, the clouds, the weather systems,

those were outside of mathematics.

NARRATOR: Until the 1970s,

when Benoit Mandelbrot introduced his new geometry.

DEVLIN: Mandelbrot came along and said "Hey, guys, all you need to do

"is look at these patterns of nature in the right way,

"and you can apply mathematics.

"There is an order beneath the seeming chaos.

"You can write down formulas

"that describe clouds, and flowers and plants.

"It's just that they're different kinds of formulas,

and they give you a different kind of geometry."

The big question is why did it take till the 1970s

Before somebody wrote a book

called The Fractal Geometry of Nature.

If they're all around us,

why didn't we see them before?

The answer seems to be

well, people were seeing them before.

People clearly recognized this repeating quality in nature.

NARRATOR: People like the great 19th century Japanese artist

Katsushika Hokusai.

If you look well enough, you see a shadow of a cloud

over Mount Fuji.

The cloud is billows upon billows upon billows.

TAYLOR: Hokusai, the great wave.

You know, on top of the great wave,

there's smaller waves.

MANDELBROT: After my book

mentioned that Hokusai was fractal,

I got inundated with people saying,

"Now we understand Hokusai."

Hokusai was drawing fractals.

TAYLOR: Everybody thinks

that mathematicians are very different from artists.

I've come to realize that art

is actually really close to mathematics,

and that they're just using different language.

And so, for Mandelbrot, it's not about equations.

It's about how do we explain this visual phenomenon.

NARRATOR: Mandelbrot's fascination

with the visual side of math began when he was a student.

MANDELBROT: It is only in January '44 that suddenly,

I fell in love with mathematics--

and not mathematics in general--

with geometry in its most concrete, sensual form.

That part of geometry which...

in which mathematics and the eye meet.

The professor was talking about algebra,

but I began to see in my mind geometric pictures which fitted

this algebra, and once you see these pictures,

the answer become obvious.

So, I discovered something which I had no clue before,

that I knew how to transform in my mind instantly

the formulas into pictures.

NARRATOR: As a young man,

Mandelbrot developed a strong sense of self-reliance,

shaped in large part

by his experience as a Jew

living under Nazi occupation in France.

For four years,

he managed to evade the constant threat

of arrest and deportation.

MANDELBROT: There is nothing more, um, hardening,

in a certain sense, than surviving a war.

Even not a soldier, but as a hunted civilian.

I knew... I knew how to act,

and I didn't trust people's wisdom very much.

NARRATOR: After the war, Mandelbrot got his Ph.D.

He tried teaching at a French university,

but he didn't seem to fit in.

MANDELBROT: They say, well,

I'm very gifted, but very misled,

and I do things the wrong way.

I was very much, um, a fish out of water.

So I abandoned this job in France and took the gamble

to go to IBM.

NARRATOR: It was 1958.

The giant American corporation was pioneering a technology

that would soon revolutionize

the way we all live:

the computer.

IBM was looking for creative thinkers--

non-conformists, even rebels.

People like Benoit Mandelbrot.

MANDELBROT: In fact, they had cornered the market

for a certain type of oddball.

We never had the slightest feeling

of being the establishment.

NARRATOR: Mandelbrot's colleagues told the young mathematician

about a problem of great concern to the company.

IBM engineers were transmitting computer data over phone lines,

but sometimes, the information was not getting through.

MANDELBROT: They realized

that every so often,

the lines became, uh, extremely noisy.

Errors occurred in large numbers.

It was indeed an extremely messy situation.

NARRATOR: Mandelbrot graphed the noise data,

and what he saw surprised him.

Regardless of the timescale, the graph looked similar.

One day:

one hour, one second--

it didn't matter.

It looked about the same.

It turned out to be self-similar with a vengeance.

NARRATOR: Mandelbrot was amazed.

The strange pattern reminded him of something

that had intrigued him as a young man--

a mathematical mystery

that dated back nearly 100 years:

the mystery of the monsters.

The story really begins in the late 19th century.

Mathematicians had written down a formal description

of what a curve must be.

But within that description, there were these other things,

things that satisfied the formal definition of what a curve is,

but were so weird that you could never draw them,

or you couldn't even imagine drawing them.

They were just regarded as monsters

or things beyond the realm.

ABRAHAM: They're not lines.

They're nothing like lines.

They're not circles.

They were, like, really, really weird.

NARRATOR: The German mathematician Georg Cantor created the first

of the monsters in 1883.

RON EGLASH: He just took a straight line, and he said,

"I'm gonna break this line into thirds,

and the middle third I'm gonna erase."

So you're left with two lines at each end.

And now I'm gonna take those two lines,

take out the middle third, and we'll do it again.

So he does that over and over again.

Most people would think,

well, if I've thrown everything away,

eventually, there's nothing left.

Not the case.

There's not just one point left.

There's not just two points left.

There's infinitely many points left.

NARRATOR: As you zoom in on the Cantor set,

the pattern stays the same,

much like the noise patterns that Mandelbrot had seen at IBM.

Another strange shape was put forward

by the Swedish mathematician Helge Von Koch.

Koch said, well, you start with an equilateral triangle,

one of the classical Euclidean geometric figures,

and on each side...

...I take a piece, and I substitute two pieces

that are now longer than the original piece.

And for each of those pieces, I substitute two pieces

that are each longer than the original piece.

Over and over again.

You get the same shape, but now,

each line has that little triangular bump on it.

And I break it again,

and I break it again, and I break it again,

and each time I break it, the line gets longer.

Every iteration, every cycle,

he's adding on another little triangle.

Imagine iterating that process of adding little bits,

infinitely many times.

What you end up with is something

that's infinitely long.

NARRATOR: The Koch Curve was a paradox.

To the eye, the curve appears to be perfectly finite.

But mathematically, it is infinite,

which means it cannot be measured.

EGLASH: At the time they called it a pathological curve,

because it made no sense, according to the way

people were thinking about measurement,

and Euclidean geometry and so on.

NARRATOR: But the Koch Curve turned out to be crucial

to a nagging measurement problem:

the length of a coastline.

In the 1940s, British scientist Lewis Richardson had observed

that there can be great variation

between different measurements of a coastline.

It depends on how long your yardstick is

and how much patience you have.

If you measure the coastline of Britain

with a one-mile yardstick, you'd get so many yardsticks,

which gives you so many miles.

If you measure it with a one-foot yardstick,

it turns out that it's longer.

And every time you use a shorter yardstick,

you get a longer number.

DEVLIN: Because you can always find finer indentations.

NARRATOR: Mandelbrot saw that the finer and finer indentations

in the Koch Curve were precisely what was needed

to model coastlines.

He wrote a very famous article in Science Magazine called

"How Long Is the Coastline of Britain?"

NARRATOR: A coastline, in geometric terms, said Mandelbrot, is a fractal.

And though he knew he couldn't measure its length,

he suspected he could measure something else: its roughness.

To do that required rethinking one of the basic concepts

in math: dimension.

What we would think of as normal geometry--

one dimension is the straight line,

two dimensions is, say, the box that has surface area.

NARRATOR: And three dimensions is a cube.

But could something have a dimension

somewhere in between, say, two and three?

Mandelbrot said, yes, fractals do.

And the rougher they are,

the higher their fractal dimension.

DEVLIN: There are all of these

technical terms, like fractal dimension,

and self-similarity,

but those are the nuts and bolts of the mathematics itself.

What that fractal geometry does is give us a way of looking at--

in a way that's extremely precise--

the world in which we live, in particular, the living world.

NARRATOR: Mandelbrot's fresh ways of thinking

were made possible by his enthusiastic embrace

of new technology.

Computers made it easy for Mandelbrot to do iteration--

the endlessly repeating cycles of calculation

that were demanded by the mathematical monsters.

MANDELBROT: The computer was totally essential.

Otherwise, it would have taken a very big, long effort.

NARRATOR: Mandelbrot decided to zero in on yet another of the monsters--

a problem introduced during World War I

by a young French mathematician named Gaston Julia.

DEVLIN: Gaston Julia--

he was actually looking at what happens when you take

a simple equation

and you iterate it through a feedback loop.

That means you take a number,

you plug it into the formula, you get a number out.

You take that number, back to the beginning,

and you feed it into

the same formula, get another number out.

And you keep iterating that over and over again.

And the question is, what happens

when you iterate it lots of times.

NARRATOR: The series of numbers you get is called a set-- the Julia set.

But working by hand,

you could never really know

what the complete set looked like.

ABRAHAM: There were attempts to draw it.

Doing a bunch of arithmetic by hand

and putting a point on graph paper.

You would have to feed it back hundreds, thousands,

millions of times.

The development of that new kind of mathematics had to wait

until fast computers were invented.

NARRATOR: At IBM, Mandelbrot did something

Julia could never do:

use a computer to run the equations millions of times.

He then turned the numbers

from his Julia sets into points on a graph.

MANDELBROT: My first step was to just draw mindlessly

a large number of Julia sets.

Not one picture, hundreds of pictures.

NARRATOR: Those images led Mandelbrot to a breakthrough.

In 1980, he created an equation of his own,

one that combined all of the Julia sets

into a single image.

When Mandelbrot iterated his equation,

he got his own set of numbers.

Graphed on a computer, it was a kind of road map

of all the Julia sets and quickly became famous

as the emblem of fractal geometry...

the Mandelbrot set.

They intersect at certain areas, and it's got like a, you know...

And they have little curlicues built into them.

Black beetle-like thing.

Crawling across the floor.

Seahorses. Dragons.

Something similar to my hair, actually.

(laughing)

NARRATOR: With this mysterious image,

Mandelbrot was issuing a bold challenge

to long-standing ideas about the limits of mathematics.

The blinders came off, and people could see forms

that were always there, but formerly were invisible.

DEVLIN: The Mandelbrot set was a great example

of what you could do in fractal geometry,

just as the archetypical example

of classical geometry is the circle.

ABRAHAM: When you zoom in, you see them coming up again,

so you see self-similarity.

You see, by zooming in, you zoom, zoom, zoom,

you're zooming in, you're zooming in,

and pop, suddenly it seems like you're exactly

where you were before, but you're not.

It's just that way down there, it has the same kind

of structure as way up here, and the sameness can be grokked.

NARRATOR: Mandelbrot's mesmerizing images

launched a fad in the world of popular culture.

MANDELBROT: Suddenly, this thing caught

like... like a bush fire.

Everybody wanted to have it.

DEVLIN: I thought, this is something big going on.

This was a cultural event of great proportions.

NARRATOR: In the late 1970s, Jhane Barnes

had just launched a business designing men's clothing.

JHANE BARNES: When I started my business in '76,

I was doing fabrics the old-fashioned way,

just on graph paper;

weaving them on a little handloom.

NARRATOR: But then, she discovered fractals

and realized that the simple rules that made them

could be used to create intricate clothing designs.

BARNES: I thought, this is amazing, so that very simple concept,

I said, "Oh, I can make designs with that."

But in the '80s, I really didn't know

how to design a fractal, because there wasn't software.

NARRATOR: So Barnes got help

from two people who knew a lot

about math and computers: Bill Jones

and Dana Cartwright.

BARNES: I had Dana and Bill writing my software for me.

They said, "Oh, your work is very mathematical."

And I was like, "It is?

That's my weakest subject in school."

We had a physicist and a mathematician

and a textile designer.

BARNES: We had so much to learn from each other.

DANA CARTWRIGHT: I did not know what a warp and a weft is.

You know, Jhane...

her ability with numbers is fairly restricted,

if I can put that politely.

All, um, the parameters here...

BARNES: There was a way we were going to communicate.

We were going to get together somehow,

and it really did happen pretty quickly.

The general fashion press thought "Jhane's a little nuts."

They started calling me the Fashion Nerd,

you know, but that was okay.

That was okay with me because I was learning a lot.

This was fun and very, very inspirational.

I'm getting things that wouldn't be possible by hand.

You know, sometimes when I think about things in my head

and I say, "You know, I just saw light coming

"through that screen door,

"and look at the moir,ing effects

that are happening on the ground."

Can I go draw that?

No way, but I can describe that to my mathematician.

This kind of reminds me of...

He sends me back the generator, all ready for me to try,

and I sit down at the computer and say,

"well let's see what it's doing."

And I have parameters that I can control.

And I keep pushing, and I go,

"well, this is not what I expected at all...

um, at all, but it's cool."

(weapons blasting)

OBI-WAN KENOBI: Use the Force, Luke.

(weapons blasting)

NARRATOR: The same kinds of fractal design principles

have completely transformed the magic of special effects.

DAN PIPONI: This is a key moment from Star Wars: Episode III,

where our two heroes have run out

onto the end of this giant mechanical arm

and the lava splashes down onto the arm.

My starting point here is to actually take

the three-dimensional model and take essentially a jet

and just shoot lava up into the air.

This looks kind of boring.

It's doing roughly the right thing,

but the motion has no kind of visual interest to it.

Let's look at what happens here

when I add the fractal swirl to it.

Where this becomes fractal is,

we take that same swirl pattern,

we shrink it down and reapply it.

We take that, we shrink it down again, we reapply it.

We shrink it down again, we reapply it.

And from here on, it's just a case

of layering up more and more and more.

I've used the same technique

to create these additional lava streams.

I then do it again here

to get some just red hot embers.

Then, we take all of those layers, and we add them up,

and we get the final composite image.

My hero lava in the foreground,

the extra lava in the background.

The embers, sparks, steam, smoke.

(grunting)

NARRATOR: Designers and artists the world over

have embraced the visual potential of fractals,

but when the Mandelbrot set was first published,

mathematicians, for the most part,

reacted with scorn.

ABRAHAM: In the Mathematical Intelligencer,

which is a gossip sheet for professional mathematicians,

there were article after article

saying he wasn't a mathematician;

he was a bad mathematician; it's not mathematics;

fractal geometry is worthless.

The eye had been banished out of science.

The eye had been excommunicated.

ABRAHAM: His colleagues, especially the really good ones,

pure mathematicians that he respected,

they turned against him.

Because, see now, you get used to the world

that you've created and that you live in,

and mathematicians had become very used

to this world of smooth curves that they could do things with.

ABRAHAM: They were clinging to the old paradigm

when Mandelbrot and a few people

were way out there bringing in the new paradigm.

And he used to call me up on the telephone late at night,

because he was bothered, and we'd talk about it.

Mandelbrot was saying,

"This is a branch of geometry just like Euclid."

Well, that offended them.

They said, "No, this is an artifact

of your stupid computing machine."

MANDELBROT: I know very well that there is this line

that fractals are pretty pictures,

but are pretty useless.

Well, it's a pretty jingle,

but it's completely ridiculous.

NARRATOR: Mandelbrot replied to his critics

with his new book: The Fractal Geometry of Nature.

It was filled with examples

of how his ideas could be useful to science.

Mandelbrot argued that with fractals,

he could precisely measure natural shapes

and make calculations that could be applied

to all kinds of formations,

from the drainage patterns of rivers

to the movements of clouds.

DEVLIN: So this domain of growing, living systems,

which I, along with most other mathematicians,

had always regarded as pretty well off-limits

for mathematics and certainly off-limits for geometry,

suddenly was center stage.

It was Mandelbrot's book that convinced us

that this wasn't just artwork.

This was new science in the making.

This was a completely new way of looking

at the world in which we live

that allowed us not just to look at it,

not just to measure it,

but to do mathematics and thereby understand it

in a deeper way than we had before.

As someone who's been working with fractals for 20 years,

I'm not going to tell you fractals are cool.

I'm going to tell you fractals are useful,

and that's what's important to me.

NARRATOR: In the 1990s, a Boston radio astronomer

named Nathan Cohen used fractal mathematics

to make a technological breakthrough

in electronic communication.

:( beeping)

Cohen had a hobby: he was a ham radio operator,

but his landlord had a rule

against rigging antennas on the building.

NATHAN COHEN: I was at an astronomy conference in Hungary,

and Dr. Mandelbrot was giving a talk

about the large-scale structure of the universe

and reporting how using fractals is a very good way

of understanding that kind of structure,

which really wowed the entire group of astronomers.

He showed several different fractals

that I, in my own mind, looked at and said,

"Oh, wouldn't it be funny

"if you made an antenna out of that shape?

I wonder what it would do."

NARRATOR: One of the first designs he tried

was inspired by one of the 19th century "monsters":

the snowflake of Helge von Koch.

I thought back to the lecture and said,

"well, I've got a piece of wire.

What happens if I bend it?"

After I bent the wire, I hooked it up

to the cable and my ham radio,

and I was quite surprised to see that it worked

the first time out of the box.

It worked very well, and I discovered

that, much of a surprise to me,

that I could actually make the antenna

much smaller using the fractal design,

so it was, frankly, an interesting way

to beat a bad rap with the landlord.

NARRATOR: Cohen's experiments soon led him to another discovery.

Using a fractal design not only made antennas smaller,

but enabled them to receive

a much wider range of frequencies.

COHEN: Using fractals, experimentally I came up

with a very wideband antenna.

And then I worked backwards

and said, "Why is it working this way?

"What is it about nature that requires you

to use the fractal to get there?"

The result of that work was

a mathematical theorem that showed

if you want to get something

that works as an antenna

over a very wide range of frequencies,

you need to have self-similarity.

It has to be fractal in its shape to make it work.

Now, that was an exact solution. It wasn't like,

"Oh, here's a way of doing it

and there's a lot of other ways of doing it."

It turned out mathematically,

we were able to demonstrate that was the only technique

you would use to get there.

(cell phone ringing)

NARRATOR: Cohen made his discovery

at a time when cell phone companies were facing a problem.

They were offering new features to their customers,

like Bluetooth, walkie-talkie, and Wi-Fi,

but each of them ran on a separate frequency.

COHEN: You need to be able to use all

those different frequencies and have access to them

without ten stubby antennas sticking out at the same time.

The alternative option is

you can let your cell phone look like a porcupine.

But most people don't want to carry around a porcupine.

NARRATOR: Today, fractal antennas are used

in tens of millions of cell phones,

and other wireless communication devices all over the world.

COHEN: We're going to see over the next ten to 15 to 20 years that

you're going to have to use fractals

because it's the only way to get, uh, cheaper costs

and smaller size for all the complex

telecommunication needs we're having.

MANDELBROT: Once you realize that

a shrewd engineer would use fractals in many, many contexts,

you better understand why nature, which is shrewder,

uses them in its ways.

They're all over in biology.

They're solutions

that natural selection has come up with

over and over and over and over again.

NARRATOR: One powerful example:

the rhythms of the heart. (beating)

Something that Boston cardiologist Ary Goldberger

has been studying his entire professional life.

ARY GOLDBERGER: The notion of sort of the human body

as a machine goes back through the tradition

of Newton and the machinelike universe.

So somehow we're, we're machines,

we're mechanisms; the heartbeat is this timekeeper.

Galileo was reported to have used

his pulse to time

the swinging of a pendular motion.

So that all fit in with the idea that a normal heartbeat

is like a metronome.

NARRATOR: But when Goldberger and his colleagues

analyzed data from thousands of people,

they found the old theory was wrong.

MADALENA DAMASIO COSTA: This is, um,

where I show the heartbeat time series of a healthy subject.

And as you can see,

the heartbeat is not constant over time.

It fluctuates, and it fluctuates a lot.

For example, in this case it fluctuates between

60 beats per minute and 120 beats per minute.

NARRATOR: The patterns looked familiar to Goldberger,

who happened to have read Benoit Mandelbrot's book.

GOLDBERGER: When you actually plotted out the intervals

between heartbeats, what you saw was very close

to the rough edges of the mountain ranges

that were in Mandelbrot's book.

You blow them up, uh, expand them,

you actually see that there are more of these

wrinkles upon wrinkles.

The healthy heartbeat, it turned out,

had this fractal architecture.

People said, "This isn't cardiology.

Do cardiology if you want to get funded."

But it turns out it is cardiology.

NARRATOR: Goldberger found that the healthy heartbeat

has a distinctive fractal pattern,

a signature that one day may help cardiologists

spot heart problems sooner.

Please look around the screen for me.

All right, Cooper, we're going to do

the calibration.

NARRATOR: At the University of Oregon,

Richard Taylor is using fractals to reveal

the secrets of another part of the body: the eye.

TAYLOR: What we want to do

is see what is that eye doing

that allows it to absorb so much visual information.

And so that's what led us into the eye trajectories.

Under the monitor is a little infrared camera,

which will actually monitor

where the eye is looking.

And it actually records that data.

And so what we get out is a trajectory

of where the eye has been looking.

Oh, it's interesting how they go around

in the patterns...

TAYLOR: And so the computer will get out this graph,

and it will look, you know, have all of these various,

uh, little structure in it.

And it's that pattern that we zoom in--

we tell the computer to zoom in on--

and, and see the fractal dimension.

NARRATOR: The tests show that the eye does not always look at things

in an orderly or smooth way.

If we could understand more about how the eye

takes in information, we could do

a better job of designing the things

that we really need to see.

TAYLOR: A traffic light.

You're looking at the traffic light.

You've got traffic.

You've got pedestrians.

Your eye is looking all over the place

trying to assess all of this information.

People design aircraft cockpits, rows of dials

and things like that.

If your eye is darting around

based on a fractal geometry, though,

maybe that's not the best way.

Maybe you don't want these things in a simple row.

We're trying to work out the natural way

that the eye wants to absorb the information.

Is it going to be similar

to a lot of these other subconscious processes?

Body motion, when you're balancing,

what are you actually doing there?

It's something subconscious, and it works.

And you're stringing together big sways

and small sways and smaller sways.

Could those all be connected together

to actually be doing a fractal pattern there?

More and more physiological processes

have been found to be fractal.

NARRATOR: Not everyone in science is convinced

of fractal geometry's potential for delivering new knowledge.

Skeptics argue that it's done little

to advance mathematical theory.

But in Toronto, biophysicist Peter Burns

and his colleagues

see fractals as a practical tool, a way to develop

mathematical models that might help

in diagnosing cancer earlier.

Detecting very small tumors is one of the big challenges

in medical imaging.

NARRATOR: Burns knew that one

early sign of cancer is particularly

difficult to see: a network

of tiny blood vessels that forms with the tumor.

Conventional imaging techniques,

like ultrasound, aren't powerful enough to show them.

BURNS: We need to be able to see structures which are

just a few tenths of a millionths of a meter across.

When it comes to a living patient,

we don't have the tools to be able

to see these tiny blood vessels.

NARRATOR: But ultrasound does provide a very good picture

of the overall movement of blood.

"Is there any way," Burns wondered,

"that images of blood flow could reveal the hidden

structure of the blood vessels?"

To find out, Burns and his colleagues

used fractal geometry

to make a mathematical model.

BURNS: If you have a mathematical way

of analyzing a structure,

you can make a model.

What fractals do is they give you some simple rules

by which you can create models.

And by changing some of the parameters of the model,

we can change how the structure looks.

NARRATOR: The model showed

the flow of blood in a kidney,

first through normal blood vessels,

and then through vessels feeding a cancerous tumor.

Burns discovered that the two kinds of networks

had very different fractal dimensions.

Instead of being neatly bifurcating,

looking like a, a nice elm tree,

the tumor vasculature is chaotic and tangled

and disorganized, looking more like a mistletoe bush.

NARRATOR: And the flow of blood

through these tangled vessels looked very different

than in a normal network-- a difference

doctors might one day be able to detect with ultrasound.

We always thought that we have to make medical images

sharper and sharper, ever more precise,

ever more microscopic in their resolution,

to find out the information

about the structure that's there.

What's exciting about this

is it's giving us microscopic information

without us actually having to look through a microscope.

We think that this fractal approach may be helpful

in distinguishing benign from malignant lesions

in a way that hasn't been possible up to now.

NARRATOR: It may take years before fractals

can help doctors predict cancer.

But they are already offering clues to one of biology's

more tantalizing mysteries:

why big animals use energy more efficiently

than little ones.

That's a question that fascinates

biologists James Brown

and Brian Enquist and physicist Geoffrey West.

GEOFFREY WEST: There is an extraordinary

economy of scale as you increase in size.

NARRATOR: An elephant, for example,

is 200,000 times heavier than a mouse,

but uses only about 10,000 times more energy

in the form of calories it consumes.

WEST: The bigger you are, you actually need less energy

per gram of tissue to stay alive.

That is an amazing fact.

NARRATOR: And even more amazing is the fact

that this relationship between the mass and energy use

of any living thing is governed by a strict

mathematical formula.

JAMES BROWN: So far as we know, that law

is universal, or almost universal, across all of life.

So it operates from the tiniest bacteria

to whales and Sequoia trees.

NARRATOR: But even though this law

had been discovered back in the 1930s,

no one had been able to explain it.

BROWN: We had this idea that

it probably had something to do with how resources

are distributed within the bodies of organisms

as they varied in size.

We took this big leap and said,

"All of life in some way

"is sustained by these underlying

"networks that are transporting oxygen,

resources, metabolites that are feeding cells."

Circulatory systems

and respiratory systems

and renal systems

and neural systems.

It was obvious that fractals were staring us in the face.

NARRATOR: If all these biological networks are fractal,

it means they obey some simple mathematical rules,

which can lead to new insights into how they work.

BROWN: If you think about it for a minute,

it would be incredibly inefficient

to have a set of blueprints

for every single stage of increasing size.

But if you have a fractal code,

a code that says when to branch

as you get bigger and bigger, then, uh,

a very simple genetic code can produce

what looks like a complicated organism.

Evolution by natural selection has hit upon a design

that appears to give the most bang for the buck.

NARRATOR: In 1997, West, Brown

and Enquist announced their controversial theory

that fractals hold the key to the mysterious relationship

between mass and energy use in animals.

Now, they are putting their theory to a bold new test:

an experiment to help determine

if the fractal structure of a single tree

can predict how an entire rain forest works.

Measurements of its trunk...

NARRATOR: Enquist has traveled to Costa Rica--

to Guanacaste province,

in the northwestern part of the country.

The government has set aside more than 300,000 acres

in Guanacaste as a conservation area.

This rain forest, like others around the world,

plays a vital role in regulating the earth's climate,

by removing carbon dioxide from the atmosphere.

If you look at the forest, it basically breathes.

And if we understand the total amount of carbon dioxide,

that's coming into, uh, these trees within this forest

we can then better understand how, uh,

this forest then ultimately regulates the total amount

of carbon dioxide in our atmosphere.

NARRATOR: With carbon dioxide levels around the world rising,

how much (:02 can rain forests like this one absorb,

and how important is their role in protecting us

from further global warming?

Enquist and a team of US scientists

think that fractal geometry may help answer these questions.

...baseline.

Let's try to get the height of the tree measured.

NARRATOR: They are going to start by doing

just about the last thing

you'd think a scientist would do here: cut down a balsa tree.

It's dying anyway,

and they have the permission of the authorities.

So Christina,

as soon as you know the height of that tree,

we can actually figure out the approximate angle

that we need to take it down on.

NARRATOR: Hooking a guide line on a high branch

helps insure the tree will land where they want it to.

Yay!

Good work.

Very good.

Very nice.

(mechanical whirring)

Nice.

Well done.

Jose, perfecto! \AIsta bien?

NARRATOR: Enquist and his colleagues

then measure the width and length

of the branches to quantify the tree's fractal structure.

Eight.

10.06.

No, that's eight.

6.3. .03.

6.0.

Eight.

Seven on the nose.

NARRATOR: They also measure how much carbon a single leaf contains,

which should allow them to figure out

what the whole tree can absorb.

So if we know the amount of carbon dioxide

that one leaf is able to take in,

then hopefully using the fractal branching rule

we can know how much carbon dioxide

the entire tree is taking in.

NARRATOR: Their next step is to move

from the tree to the whole forest.

All right, this is good.

13.2.

3.3.

ENQUIST: We're going to census this forest.

We're going to be measuring

the diameter at the base of the tree,

ranging all the way from the largest trees down

to the smallest trees.

And in that way we can then sample the distribution

of sizes within the forest.

It's 61.8 centimeters.

Even though the forest may appear random and chaotic,

the team believes it actually has a structure--

one that amazingly is almost identical

to the fractal structure of the tree they have just cut down.

BROWN: The beautiful thing is

that the distribution of the sizes

of individual trees in the forest

appears to exactly match the distribution

of the sizes of individual branches

within a single tree.

NARRATOR: If they're correct,

studying a single tree will make it easier

to predict how much carbon dioxide

an entire forest can absorb.

When they finish here,

they take their measurements back to base camp,

where they'll see if their ideas hold up.

So is this the... this is the tree plot, right?

The cool thing is that,

if you look at the tree, you see the same pattern

amongst the branches as we see amongst the trunks

in the forest. Very nice.

NARRATOR: Just as they'd predicted,

the relative number of big and small trees

closely matches the relative number

of big and small branches.

ENQUIST: It's actually phenomenal that it is parallel.

The slope of that line for the tree appears

to be the same for the forest as well.

So I guess it was worth cutting up the tree.

It was definitely worth cutting up the tree.

NARRATOR: So far, the measurements from the field

appear to support the scientist's theory

that a single tree can help scientists assess

how much this rain forest is helping

to slow down global warming.

By analyzing the fractal patterns within the forest,

that then enables us to do something

that we haven't really been able to do before.

Have then a mathematical basis

to then predict how the forest as a whole

takes in carbon dioxide and, ultimately,

that's important for understanding

what may happen with global climate change.

NARRATOR: For generations,

scientists believed that the wildness of nature

could not be defined by mathematics.

But fractal geometry is leading to a whole new understanding,

revealing an underlying order

governed by simple mathematical rules.

What I thought of in my hikes through forests,

that, you know, it's just a bunch of trees

of different sizes, big ones here, small ones there,

looking like it's sort of some arbitrary chaotic mess

actually has an extraordinary structure.

NARRATOR: A structure that can be mapped out

and measured using fractal geometry.

ENQUIST: What's absolutely amazing is that you can

translate what you see in the natural world

in the language of mathematics.

And I can't think of anything more beautiful than that.

Math is our one and only strategy

for understanding the complexity of nature.

Now, fractal geometry has given us

a much larger vocabulary.

And with the larger vocabulary

we can read more of the book of nature.

On NOVA's "Hidden Dimension" Web site,

explore the Mandelbrot set, see a gallery of fractal images

and much more.

Find it on pbs.org.

Major funding for NOVA is provided by the following:

Taking on the world's toughest energy challenges.

And by:

And...

And by the Corporation for Public Broadcasting

and by contributions to your PBS station from:

Captioned by Media Access Group at WGBH access.wgbh.org