Fractals Documentary

[Cripin Plascencia]

This program not available for streaming.) 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.

NARRATOR: You can find it in the rain forest, on thefrontiers of medical research, in the movies, and it's all over the world ofwireless communications. One of nature's biggest design secrets has finally beenrevealed.

Well,it might surprise a lot of people that ExxonMobil would be interested inlithium ion battery technology applied to hybrid electric vehicles. Our newbattery separator film is a true breakthrough that's going to enable thedeployment of more hybrid vehicles, faster. This means a tremendous reductionin greenhouse gases, the equivalent of removing millions of cars from the road.I think this is the most important project that I've worked on in my career.

LORENCARPENTER: I would get the data from themand make pictures from various angles, but I wanted to be able to put amountain behind it, because every Boeing publicity photo in existence has amountain behind it. But there was no way to do mountains. Mountains hadmillions and millions of little triangles or polygons or whatever you want tocall it, and we had enough trouble with a hundred. Especially in those dayswhen our machines were slower than the ones you have in your watch.

LORENCARPENTER: In 1978, I ran into this bookin a bookstore: Fractals: Form, Chance and Dimension, by BenoitMandelbrot, and it has to do with the fractal geometry of nature. So I boughtthe 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 innature can be described mathematically as fractals: a word he invented todefine shapes that look jagged and broken. He said that you can create afractal by taking a smooth-looking shape and breaking it into pieces, over andover again.

Themethod is dead simple. You start with a landscape made out of very roughtriangles, big ones. And then for each triangle, break it into, into fourtriangles. And then do that again, and then again and again and again.

LORENCARPENTER: The pictures were stunning. Theywere just totally stunning. No one has had ever seen anything like this. And I just opened awhole new door to a new world of making pictures. And it got the computergraphics community excited about fractals, because, suddenly, they were easy todo. And so people started doing them all over the place.

KEITHDEVLIN: The key tofractal geometry, and the thing that evaded anyone until, really, Mandelbrotsort of said, "This is the way to look at things, is that if you look on thesurface, you see complexity, and it looks very non-mathematical." WhatMandelbrot said was that..."think not of what you see, but what it took toproduce what you see."

BRIANENQUIST (University of Arizona): If we look at each of the nodes, the branchingnodes of this tree, what you'll actually see is that the pattern of branching isvery similar throughout the tree. As we go from the base of the tree to higherup, you'll see we have mother branches then branching then into daughterbranches.

Ifwe 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 stalkof broccoli, to the surface of the moon, to the arteries that transport bloodthrough our bodies. But Mandelbrot's fascination with these irregular-lookingshapes put him squarely at odds with centuries of mathematical tradition.

KEITHDEVLIN: Mandelbrotcame along and said, "Hey, guys, all you need to do is look at these patternsof nature in the right way, and you can apply mathematics. There is an orderbeneath the seeming chaos. You can write down formulas that describe clouds andflowers and plants. It's just that they're different kinds of formulas, andthey give you a different kind of geometry."

RICHARDTAYLOR (University of Oregon): The big question is, why did it take 'til the 1970sbefore 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 repeatingquality in nature.

RICHARDTAYLOR: Everybody thinks thatmathematicians are very different from artists. I've come to realize that artis actually really close to mathematics, and that they're just using differentlanguage. And so for Mandelbrot it's not about equations. It's about, "How dowe explain this visual phenomenon?"

NARRATOR: As a young man, Mandelbrot developed a strongsense of self-reliance, shaped in large part by his experience as a Jew, livingunder Nazi occupation in France. For four years, he managed to evade theconstant threat of arrest and deportation.

BENOITMANDELBROT: There is nothing morehardening, in a certain sense, than surviving a war, even, not a soldier, but as a hunted civilian. I knew how toact, and I didn't trust people's wisdom very much.

NARRATOR: Mandelbrot's colleagues told the youngmathematician about a problem of great concern to the company. IBM engineerswere transmitting computer data over phone lines, but sometimes the informationwas not getting through.

NARRATOR: Mandelbrot graphed the noise data, and what hesaw surprised him. Regardless of the timescale, the graph looked similar: oneday, one hour, one second, it didn't matter. It looked about the same.

NARRATOR: Mandelbrot was amazed. The strange patternreminded him of something that had intrigued him as a young man: a mathematicalmystery that dated back nearly a hundred years, the mystery of the "monsters."

KEITHDEVLIN: The storyreally begins in the late 19th century. Mathematicians had written down aformal description of what a curve must be. But within that description, therewere these other things, things that satisfied the formal definition of what acurve is but were so weird that you could never draw them, or you couldn't evenimagine drawing them. They were just regarded as "monsters" or "things beyondthe realm."

RONEGLASH (Rensselaer Polytechnic Institute): He just took a straight line and he said, "I'm goingto break this line into thirds, and the middle third I'm going to erase. So you'releft 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 andover again.

KEITHDEVLIN: Mostpeople would think, "Well, if I've thrown everything away, eventually there'snothing left." Not the case; there's not just one point left, there's not justtwo points left. There's infinitely many points left.

LORENCARPENTER: "...I take a piece and Isubstitute two pieces that are now longer than the original piece. And for eachof those pieces, I substitute two pieces that are each longer than the originalpiece..."

LORENCARPENTER: It depends on how long youryardstick is and how much patience you have. If you measure the coastline ofBritain with a one-mile yardstick, you'd get so many yardsticks, which givesyou so many miles. If you measure it with a one-foot yardstick, it turns outthat it's longer. And every time you use a shorter yardstick, you get a longernumber.

NARRATOR: A coastline, in geometric terms, said Mandelbrot,is a fractal. And though he knew he couldn't measure its length, he suspectedhe could measure something else: its roughness. To do that required rethinkingone of the most basic concepts in math: dimension.

NARRATOR: And three dimensions is a cube. But couldsomething have a dimension somewhere in between, say, two and three? Mandelbrotsaid, "Yes. Fractals do. And the rougher they are, the higher their fractaldimension."

NARRATOR: Mandelbrot's fresh ways of thinking were madepossible by his enthusiastic embrace of new technology. Computers made it easyfor Mandelbrot to do iteration, the endlessly repeating cycles of calculationthat were demanded by the mathematical monsters.

KEITHDEVLIN: GastonJulia, he was actually looking at what happens when you take a simple equationand you iterate it through a feedback loop. That means you take a number, youplug it into the formula, you get a number out. You take that number, goback to the beginning, and you feed itinto the same formula, get another number out. And you keep iterating that overand over again.

WhenMandelbrot iterated his equation he got his own set of numbers. Graphed on acomputer, it was a kind of roadmap of all the Julia sets and quickly becamefamous as the emblem of fractal geometry: the Mandelbrot set.

RALPHABRAHAM: When you zoom in, you see themcoming 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 itseems like you're exactly where you were before, but you're not. It's just thatway down there it has the same kind of structure as way up here. And thesameness can be grokked.

JHANEBARNES: I thought, "This isamazing." So, that very simple concept, I said, "Oh, I can make designs withthat." But in the '80s, I really didn't know how to design a fractal becausethere wasn't software.

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