All language subtitles for Fractals Hunting the Hidden Dimension 1080p PBS NOVA 2008 EN SUB

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."

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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
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Group at WGBH access.wgbh.org