Perspective
Two eyes can make sense of line, shape… often color… sometimes texture, motion. . .
A mind’s eye can visualize the future… the imagined… the intended.
We can see things that aren’t there.
Many times we can’t see things that are there.
Until someone tells us they are.
Then, multiple eyes seeing things with new focus can do so much more.
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“It is the obvious which is so difficult to see most of the time. People say 'It's as plain as the nose on your face.' But how much of the nose on your face can you see, unless someone holds a mirror up to you?”
― Isaac Asimov, I, Robot
"Sometimes all it takes is a tiny shift of perspective to see something familiar in a totally new light."
— Dan Brown, The Lost Symbol
This is one of those stories...
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 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.
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."
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 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. 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.
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.
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 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.
PBS Airdate: October 28, 2008
Find the full transcript here
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 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 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.
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: From the time movies began, animators had to draw each frame by hand—thousands of them—to make even a short cartoon.
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 references, twice.
NARRATOR: In his book, Mandelbrot said that many forms in nature can be described mathematically as fractals: a word he invented to define shapes that look jagged and broken. He said that you can create a fractal by taking a smooth-looking shape and breaking it into pieces, over and over again.
LOREN CARPENTER: Within three days, I was producing pictures of mountains on my computer at work. The pictures were stunning. They were just totally stunning. No one 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 computer-generated sequence in a feature film 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.
KEITH DEVLIN: What Mandelbrot said was that..."think not of what you see, but what it took to produce what you see."
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.
. . . 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 investigation.
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 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."
Hunting the Hidden Dimension