Hunting the Hidden Dimension
Mysteriously beautiful fractals are shaking up the world of mathematics and deepening our understanding of nature. Airing August 24, 2011 at 9 pm on PBS Aired August 24, 2011 on PBS
Program Description
The producers wish to acknowledge the contributions to the film of Dr. Wolfgang Beyer, who created 12 of the images of the Mandelbrot set which we used in the film. We apologize for inadvertently omitting Dr. Beyer's credit from the film itself. Dr. Beyer's images, which were created with the software program Ultra Fractal 3, can be found at http://commons.wikimedia.org/wiki/User:Wolfgangbeyer. For more information, visit www.wolfgangbeyer.de.
Transcript
Hunting the Hidden Dimension
PBS Airdate: October 28, 2008
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 revealed.
GEOFFREY WEST (Santa Fe Institute): My god, of course. It's obvious.
NARRATOR: It's an oddlooking 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 again.
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 centuriesold 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 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.
Major funding for NOVA is provided by the following:
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Discover new knowledge: HHMI.
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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NARRATOR: In 1978, at Boeing Aircraft in Seattle, engineers were designing experimental aircraft.
LOREN CARPENTER (Pixar Animation Studios): Exotic things, with two wings or two tails or two fuselages, and just weird stuff because, "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.
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 short cartoon.
THUMPER (Bambi/Filmclip): That's why they call me Thumper.
NARRATOR: But that was before Loren Carpenter stumbled across the work of a littleknown 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 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 smoothlooking 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 of Khan.
It was the first ever completely computergenerated sequence in a feature film...
LEONARD NIMOY (As Mr. Spock, Star Trek II: The Wrath of Khan/Filmclip): 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.
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 human body.
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 nonmathematical." 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 selfsimilarity.
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 going.
One of the most familiar examples of selfsimilarity 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 selfsimilarity 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 irregularlooking 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 investigation.
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 humanmade 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...
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 wellsuited 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.
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 geometry."
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 Hokusai.
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 drawing fractals."
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 visual phenomenon?"
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 selfreliance, 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, nonconformists, 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 getting through.
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 selfsimilar 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 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.
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 triangle.
KEITH DEVLIN: 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.
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 onemile yardstick, you'd get so many yardsticks, which gives you so many miles. If you measure it with a onefoot 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 selfsimilarity, 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.
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 Gaston Julia.
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 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.
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 single image.
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, you know...
LOREN CARPENTER: And they have little curlicues built into them.
DANA CARTWRIGHT (Designer Software LLC): ...black beetlelike 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 the circle.
RALPH ABRAHAM: When you zoom in, you see them coming up again, so you see selfsimilarity. 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 grokked.
NARRATOR: Mandelbrot's mesmerizing images launched a fad in the world of popular culture.
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 oldfashioned 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.
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 designer.
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 threedimensional 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 and more.
I've used the same technique to create these additional lava streams. I then do it again, here, to get some...just redhot 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 been excommunicated.
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 offlimits for mathematics, and certainly offlimits 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 communication.
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 largescale 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.
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 verywideband 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 selfsimilarity. 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, walkietalkie and WiFi, but each of them ran on a separate frequency.
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 we're having.
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 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 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 show them.
PETER BURNS: We need to be able to see structures which are just a few tenthsofamillionths 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 bloodflow could reveal the hidden structure of the blood vessels?
To find out, Burns and his colleagues used fractal geometry to make a mathematical model.
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 dimensions.
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 size.
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 it consumes.
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 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.
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 size.
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 work.
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 atmosphere.
NARRATOR: With carbon dioxide levels around the world rising, how much CO_{2} 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 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.
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 down on.
NARRATOR: Hooking a guide line on a high branch helps insure the tree will land where they want it to.
GROUP: Yay!
BRIAN ENQUIST: Very good, very nice.
Jose, perfecto!
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.
CATHY: Eight.
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 taking in.
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 the forest.
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 mathematical rules.
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 geometry.
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 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.
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Hunting the Hidden Dimension
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 Image credit: (yellow Mandelbrot set) Courtesy Niche Software
Participants
 Ralph Abraham
 University of California, Santa Cruz www.math.ucsc.edu/Faculty/abraham.html
 Jhane Barnes
 Jhane Barnes, Inc. www.jhanebarnes.com/
 James Brown
 University of New Mexico
 Peter Burns
 University of Toronto medbio.utoronto.ca/faculty/burns.html
 Loren Carpenter
 Pixar Animation Studios theoscarsite.com/whoswho7/carpenter_l.htm
 Dana Cartwright
 Designer Software LLC www.weavemaker.com/1/about/about.php
 Nathan Cohen
 Fractal Antenna Systems, Inc. www.fractenna.com/nca_cohen_bio.html
 Madalena Damasio Costa
 Harvard Medical School reylab.bidmc.harvard.edu/people/madalena/index.html
 Keith Devlin
 Stanford University www.stanford.edu/~kdevlin/
 Ron Eglash
 Rensselaer Polytechnic Institure www.rpi.edu/~eglash/eglash.htm
 Brian Enquist
 University of Arizona eeb37.biosci.arizona.edu/~brian/enquist_lab_folks.html
 Ary Goldberger
 Harvard Medical School reylab.bidmc.harvard.edu/people/Ary.shtml
 Christina Lamanna
 University of Arizona physics.uwyo.edu/~clamanna/
 Benoit Mandelbrot
 Yale University www.math.yale.edu/mandelbrot/
 Jennifer Ouellette
 Science Writer www.jenniferouellettewrites.com/Biography2.html
 Dan Piponi
 Industrial Light and Magic homepage.mac.com/sigfpe/
 Richard Taylor
 University of Oregon www.uoregon.edu/~msiuo/taylor/art/info.html
 Geoffrey West
 Santa Fe Institute www.santafe.edu/profiles/?pid=64
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