[program video viewable at url in five chapters]
They're odd-looking shapes you may never have heard of, but they're
everywhere around you-the jagged repeating forms called fractals. If you
know what to look for, you can find them in the clouds, in mountains, even
inside the human body.
running time 11:36
THE MANDELBROT SET
In 1958, Benoit Mandelbrot begins using computers to explore vexing problems
in math. They help him to understand repeating patterns in nature in an
entirely new way. He coins the term fractal to describe them and develops
the Mandelbrot set in 1980.
running time 9:51
ON THE DEFENSE
Though many colleagues initially scorned Mandelbrot's work, his mesmerizing
fractal images launched a popular culture fad. More importantly, his book
The Fractal Geometry of Nature explained how his ideas could be applied in
the real world. Mandelbrot's ideas inspire an ever-increasing number of
applications, including the fractal antenna.
running time 10:40
FRACTALS IN THE BODY
Fractal patterns turn up everywhere in biology, from the irregular rhythm of
the heart to basic eye function. The fractal nature of such physiological
processes, which obey simple mathematical rules, offers hope of better
diagnosis and treatment of problems as well as new insights into how such
running time 10:15
NATURE'S FRACTAL NATURE
With carbon dioxide levels around the world rising, a team of American
scientists travels to a rain forest in Costa Rica. They employ fractal
geometry to analyze how much CO2 the rain forest can absorb.
running time 7:52
You may not know it, but fractals, like the air you breathe, are all around
you. Their irregular, repeating shapes are found in cloud formations and
tree limbs, in stalks of broccoli and craggy mountain ranges, even in the
rhythm of the human heart. In this film, NOVA takes viewers on a fascinating
quest with a group of maverick mathematicians determined to decipher the
rules that govern fractal geometry.
For centuries, fractal-like irregular shapes were considered beyond the
boundaries of mathematical understanding. Now, mathematicians have finally
begun mapping this uncharted territory. Their remarkable findings are
deepening our understanding of nature and stimulating a new wave of
scientific, medical, and artistic innovation stretching from the ecology of
the rain forest to fashion design. The documentary highlights a host of
filmmakers, fashion designers, physicians, and researchers who are using
fractal geometry to innovate and inspire.
Produced and directed by Emmy- and Peabody Award-winning filmmakers Michael
Schwarz and Bill Jersey, the documentary weaves cutting-edge research from
the front lines of science into a compelling mathematical detective story.
The film introduces a number of distinguished individuals who have used
fractal geometry to transform their fields, like Loren Carpenter, who
created the first completely computer-generated sequence in a movie.
In the late 1970s, Carpenter stumbled across the work of a little-known
mathematician, Benoit Mandelbrot, who coined the word "fractal," from the
Latin word fractus, meaning irregular or broken up. Based on Mandelbrot's
mathematical descriptions of fractals in nature, Carpenter was able to
create detailed computer simulations of organic forms in a way that had
never before been possible. The groundbreaking computer-generated sequence
Carpenter produced in 1980 for Star Trek II: The Wrath of Khan marked a
milestone in movie history, and owed its creation to fractal geometry.
It took a maverick with a hard-won aversion to authority to stand up to the
conventional wisdom that nature stood outside the bounds of mathematics.
Through interviews and personal artifacts, Mandelbrot shares the story of
his struggle to survive as a Jewish teenager in Nazi-occupied France, his
journey to America, and his lifelong fascination with a corps of European
mathematicians whose explorations of the so-called "mathematical monsters"
laid the groundwork for his own discoveries. [Read an interview with
Mandelbrot, illustrated with stunning fractal images.]
Filmmaker Bill Jersey believes Mandelbrot's approach to fractals might
ultimately become as significant as the cracking of the genetic code. "As
fractals continue to revolutionize the way scientists develop theories and
conduct research, the inevitable results will be innovations that
dramatically change health care, environmental policy, design, and
technology," Jersey says.
In 1980, Mandelbrot published a mesmerizing image known as the Mandelbrot
set. (To explore the set, go to A Sense of Scale.) The intricate, mysterious
beauty of this image, which was generated by a single mathematical function,
won him acclaim from an unexpected quarter-the world of popular culture. But
fractals are more than pretty pictures. Almost all living things distribute
nutrients through their bodies via branching networks, such as systems of
blood vessels, that obey the rules of fractal geometry.
In Toronto, physicist Peter Burns is making a mathematical model of blood
vessels to find ways to diagnose cancer earlier than is now possible. In
Boston, cardiologist Ary Goldberger has discovered that, contrary to
centuries of belief, a healthy human heartbeat does not have an even pattern
like a metronome but rather a jagged, variable fractal pattern-a discovery
that one day may help doctors diagnose cardiac disease before damage is
"This film is about looking at the world around us in a completely different
way," says producer Michael Schwarz. "If you pay attention, you can see that
fractals appear throughout nature. But until Benoit Mandelbrot came along,
no one really understood what was there all along."
NARRATOR: You can find it in the 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
GEOFFREY WEST (Santa Fe Institute): 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.
JAMES BROWN (University of New Mexico): They're all over in biology. They're
solutions that natural selection has come up with over and over and over
NARRATOR: Fractals are in our lungs, kidneys and blood vessels.
KEITH DEVLIN (Stanford University): 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.
BENOIT MANDELBROT (Yale University): 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.
RALPH ABRAHAM (University of California, Santa Cruz): The blinders came off,
and people could see forms that were always there but, formerly, were
NARRATOR: In 1978, at Boeing Aircraft in Seattle, engineers were designing
LOREN CARPENTER (Pixar Animation Studios): Exotic things, with two wings or
two tails or two fuselages, and just weird stuff because, "who knows, it
NARRATOR: A young computer scientist named Loren Carpenter was helping them
visualize what the planes might look like in flight.
LOREN CARPENTER: I would get the data from them and make pictures from
various angles, but I wanted to be able to put a mountain behind it, 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 we had
enough trouble with a hundred. Especially in those days when our machines
were 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
THUMPER (Bambi/Filmclip): 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.
LOREN CARPENTER: In 1978, I ran into this book in 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 and took it home and read
it, cover to cover, every last little word, including the footnotes and
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.
LOREN 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, into four
triangles. And then do that again, and then again and again and again.
NARRATOR: Endless repetition-what mathematicians call iteration-it's one of
the keys to fractal geometry.
LOREN CARPENTER: The pictures were stunning. They were just totally
stunning. No one has had 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
It was the first ever completely computer-generated sequence in a feature
LEONARD NIMOY (As Mr. Spock, Star Trek II: The Wrath of Khan/Filmclip):
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.
BENOIT MANDELBROT: 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
KEITH DEVLIN: 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
BENOIT MANDELBROT: The main idea is always-as you zoom in and zoom out-the
object looks the same.
KEITH DEVLIN: 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
One of the most familiar examples of self-similarity is a tree.
BRIAN ENQUIST (University of Arizona): 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 have mother branches then 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.
BENOIT MANDELBROT: In the whole of science, the whole of mathematics,
smoothness was everything. What I did was to open up roughness for
KEITH DEVLIN: We used mathematics to build the pyramids, to construct the
Parthenon. We used 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
RALPH ABRAHAM: ...circles, triangles...
KEITH DEVLIN: ...flat surfaces...
RALPH ABRAHAM: ...pyramids, tetrahedrons, icosahedrons, dodecahedrons.
LOREN CARPENTER: Smooth edges.
KEITH 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
KEITH 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
RICHARD TAYLOR (University of Oregon): The big question is, why did it take
'til 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
BENOIT MANDELBROT: If you look well enough, you see a shadow of a cloud over
Mount Fuji. The cloud is billows upon billows upon billows.
RICHARD TAYLOR: Hokusai, "The Great Wave," you know, on top of the great
wave there's smaller waves.
BENOIT MANDELBROT: After my book mentioned that Hokusai was fractal, I got
inundated with people saying, "Now we understand Hokusai. Hokusai was
RICHARD 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
NARRATOR: Mandelbrot's fascination with the visual side of math began when
he was a student.
BENOIT 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 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 answers 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.
BENOIT MANDELBROT: There is nothing more hardening, in a certain sense, than
surviving a war, even, not a soldier, but as a hunted civilian. 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.
BENOIT MANDELBROT: They say, well, I'm very gifted but very misled, and I do
things the wrong way. I was very much a fish out of water. So I abandoned
this job in France and took a 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.
NARRATOR: IBM was looking for creative thinkers, non-conformists, even
rebels; people like Benoit Mandelbrot.
BENOIT 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
BENOIT MANDELBROT: They realized that every so often the lines became
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.
BENOIT MANDELBROT: 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 a hundred years, the mystery of the "monsters."
KEITH DEVLIN: 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."
RALPH 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 (Rensselaer Polytechnic Institute): He just took a straight line
and he said, "I'm going to break this line into thirds, and the middle third
I'm going to erase. So you're left with two lines at each end. And now I'm
going to take those two lines, take out the middle third, and we'll do it
again." So he does that over and over again.
KEITH DEVLIN: 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
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
KEITH DEVLIN: Koch said was, "Well, you start with an equilateral triangle,
one of the classical Euclidean geometric figures, and on each side..."
LOREN CARPENTER: "...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..."
KEITH DEVLIN: "...over and over again."
RON EGLASH: You get the same shape, but now each line has that little
triangular bump on it.
LOREN CARPENTER: 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.
RON EGLASH: Every iteration, every cycle, he's adding on another little
KEITH DEVLIN: Imagine iterating that process of adding little bits,
infinitely, many times. What you end up with is something that's infinitely
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
RON 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.
LOREN CARPENTER: 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.
KEITH 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.
KEITH DEVLIN: 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 most basic concepts in math: dimension.
JENNIFER OUELLETTE (Science Writer): 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."
KEITH 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
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.
BENOIT 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
KEITH 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, go back to the beginning, and you feed it into the
same formula, get another number out. And you keep iterating that over and
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
RALPH ABRAHAM: There were attempts to draw it, doing a bunch of arithmetic
by hand and putting a point on graph paper.
KEITH DEVLIN: 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.
BENOIT 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
When Mandelbrot iterated his equation he got his own set of numbers. Graphed
on a computer, it was a kind of roadmap of all the Julia sets and quickly
became famous as the emblem of fractal geometry: the Mandelbrot set.
JENNIFER OUELLETTE: They intersect at certain areas, and it's got like a,
LOREN CARPENTER: And they have little curlicues built into them.
DANA CARTWRIGHT (Designer Software LLC): ...black beetle-like thing...
RICHARD TAYLOR: ...crawling across the floor.
ARY GOLDBERGER (Harvard Medical School): Seahorses.
RALPH ABRAHAM: Dragons.
RICHARD TAYLOR: Something similar to my hair, actually.
NARRATOR: With this mysterious image, Mandelbrot was issuing a bold
challenge to longstanding ideas about the limits of mathematics.
RALPH ABRAHAM: The blinders came off and people could see forms that were
always there, but, formerly, were invisible.
KEITH 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
RALPH 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, and 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
NARRATOR: Mandelbrot's mesmerizing images launched a fad in the world of
BENOIT MANDELBROT: Suddenly, this thing caught like a bush fire. Everybody
wanted to have it.
KEITH 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 (Jhane Barnes, Inc.): When I started my business, in '76, I was
doing fabrics the old-fashioned way, just on graph paper, weaving them on a
NARRATOR: But then she discovered fractals and realized that the simple
rules that made them could be used to create intricate clothing designs.
JHANE 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.
JHANE 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."
DANA CARTWRIGHT: We had a physicist and a mathematician and a textile
JHANE 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.
JHANE 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, 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
moireing effects that are happening on the ground."
Can I go draw that? No way. But I can describe that to my mathematician. 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, "Wow, this is not what I
expected at all, at all." But it's cool.
MOVIE VOICE (Star Wars/Film Clip): Use the force, Luke.
NARRATOR: The same kinds of fractal design principles have completely
transformed the magic of special effects.
WILLI GEIGER (Industrial Light & Magic): 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, 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
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.
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.
RALPH 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."
BENOIT MANDELBROT: The eye had been banished out of science. The eye had
RALPH ABRAHAM: His colleagues, especially the really good ones, pure
mathematicians that he respected, they turned against him.
KEITH DEVLIN: Because, see now, you, 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.
RALPH 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."
BENOIT 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.
KEITH 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.
NATHAN COHEN (Fractal Antenna Systems, Inc.): 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
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
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.
NATHAN COHEN: 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.
NATHAN COHEN: Using fractals, experimentally, I came up with a
very-wide-band 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. And 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 we could use to get there.
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
NATHAN COHEN: You need to be able to use all those different frequencies and
have access to them, without 10 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.
NATHAN COHEN: We're going to see over the next 10 to 15 to 20 years that
you're going to have to use fractals because it's the only way to get
cheaper costs and smaller size for all the complex telecommunication needs
BENOIT MANDELBROT: Once you, 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.
JAMES BROWN: They're all over in biology. They're solutions that natural
selection has come up with over and over and over again.
NARRATOR: One powerful example: the rhythms of the heart. Something that
Boston cardiologist Ary Goldberger has been studying his entire professional
ARY GOLDBERGER: The notion of, sort of, the human body as a machine goes
back through the tradition of Newton and the machine-like universe. So
somehow 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 the 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 (Harvard Medical School): This is 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.
ARY 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, 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.
At the University of Oregon, Richard Taylor is using fractals to reveal the
secrets of another part of the body: the eye.
RICHARD 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. And so the computer will get out this graph, and it will
look...you know...have all of these various little structures in it. And
it's that pattern that we zoom in...we tell the computer to zoom in on 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.
RICHARD 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.
PETER BURNS (University of Toronto): 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
PETER 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
PETER 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
PETER BURNS: 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 those tangled vessels looked very
different than in a normal network, a difference doctors might one day be
able to detect with ultrasound.
PETER BURNS: 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
NARRATOR: An elephant, for example, is 200,000 times heavier than a mouse,
but uses only about ten thousand times more energy in the form of calories
GEOFFREY 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
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
NARRATOR: But even though this law had been discovered back in the 1930s, no
one had been able to explain it.
JAMES 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
GEOFFREY WEST: 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
JAMES 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 a very simple genetic code can produce what
looks like a complicated organism.
BRIAN ENQUIST: 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 rainforest works.
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 rainforest, like
others around the world, plays a vital role in regulating the Earth's
climate, by removing carbon dioxide from the atmosphere.
BRIAN ENQUIST: If you look at the xforest, it, basically, breathes. And if
we understand the total amount of carbon dioxide that's coming into these
trees within this forest, we can then better understand how this forest
then, ultimately, regulates the total amount of carbon dioxide in our
NARRATOR: With carbon dioxide levels around the world rising, how much CO2
can rainforests like this one absorb, and how important is their role in
protecting us from further global warming?
Enquist and a team of U.S. scientists think that fractal geometry may help
answer these questions.
BRIAN ENQUIST: ...baseline. Let's try to get the height of the tree
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.
BRIAN ENQUIST: 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
NARRATOR: Hooking a guide line on a high branch helps insure the tree will
land where they want it to.
BRIAN ENQUIST: Very good, very nice.
NARRATOR: Enquist and his colleagues then measure the width and length of
the branches to quantify the tree's fractal structure.
BRIAN ENQUIST: Eight.
CATHY: Ten point zero six.
BRIAN ENQUIST: No, that's eight. Six point three
MALE: Point zero three.
MALE: Six, zero.
MALE: 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.
CHRISTINA LAMANNA (Santa Fe Institute): 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
NARRATOR: Their next step is to move from the tree to the whole forest.
BRIAN ENQUIST: All right, this is good.
BRAD BOYLE: Thirteen point two, three point three.
BRIAN ENQUIST: We're going to census this forest. We're going to be
measuring the diameter at the base of the trees, 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.
BRAD BOYLE: It's 61.8 centimeters.
NARRATOR: 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.
JAMES 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.
BRIAN ENQUIST: So is this the, this is the tree plot, right?
DREW KERKHOFF: Yeah. 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
NARRATOR: Just as they predicted, the relative number of big and small trees
closely matches the relative number of big and small branches.
BRIAN 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.
DREW KERKHOFF: So I guess it was worth cutting up the tree.
BRIAN ENQUIST: It was definitely worth cutting up the tree.
NARRATOR: So far, the measurements from the field appear to support the
scientists' theory that a single tree can help scientists assess how much
this rainforest is helping to slow down global warming.
BRIAN ENQUIST: 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
GEOFFREY WEST: 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
BRIAN 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.
RALPH ABRAHAM: 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 larger vocabulary we can read more of the book of
Major funding for Hunting the Hidden Dimension is provided by the Alfred P.
Sloan Foundation, to portray the lives of men and women engaged in
scientific and technological pursuit.
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