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1. Fractal Geometry Yale University. Michael Frame, Benoit Mandelbrot, and Nial Neger ... This is a collection of pages meant to support a first course in fractal geometry for ... classes.yale.edu/fractals/ - Cached - Similar 2. Benoit B. Mandelbrot by B Mandelbrot - Cited by 4 - Related articles Benoit Mandelbrot. FRACTAL AND MULTIFRACTAL FINANCE Crashes and Long- Dependence. VIRTUAL SELECTA ... that are found in this home site http://.math.yale.edu/mandelbrot ... Last remark: this is not the only webbook on this website. ... www.math.yale.edu/mandelbrot/webbooks/wb_fin.html - Cached - Similar 3. [DOC] Books on fractals [DOC] - Yale Mathematics Department File Format: Microsoft Word - View as HTML FRACTAL GEOMETRY. Prepared for Benoit B. Mandelbrot (BBM) ..... Iteration: A Toolkit of Dynamics Activities • Key Curriculum Press • 1998...... B. B. MANDELBROT • Virtual books • On the website http://www.math.yale.edu/mandelbrot ... www.math.yale.edu/mandelbrot/web_docs/chronicleFracta ls.doc - Similar Show more results from www.math.yale.edu 4. Fractal Geometry by Michael Frame, Benoit Mandelbrot , Nial Neger ... Fractal Geometry by Michael Frame, Benoit Mandelbrot, Nial Neger - free book at E- Books Directory ... by Michael Frame, Benoit Mandelbrot, Nial Neger. Publisher: Yale University 2009 ... Your (nick)name (as will appear on website) ... www.e-booksdirectory.com › Science - Cached - Similar 5. [PDF] Fractal Geometry File Format: PDF/Adobe Acrobat - Quick View http://classes.yale.edu/fractals/ of Michael Frame, Benoit Mandelbrot and Nial Neger. ... Europa has a very icy fractal surface. NASA's Planetary Web Site, ... View: “Mandelbrot's World of Fractals” DVD by Key Curriculum ... www.fractalmath.conncoll.edu/materials/FracG1Revised.pdf - Similar 6. Benoit B. Mandelbrot - Fractal Wisdom Jul 13, 2010 ... Benoit B. Mandelbrot - The Story of Benoit B. Mandelbrot and the Geometry of Chaos. ... Benoit Mandelbrot, now both an IBM scientist and Professor of Mathematics at Yale, made his great discoveries by .... Benoit Mandelbrot: Fractals and the art of roughness ... Site Designed by: Business of Image ... www.fractalwisdom.com/science-of-chaos/benoit-b-mandelbrot/ - Cached 7. Macmillan: The Fractal Geometry of Nature: Benoit B. Mandelbrot : Books add this book's widget to your site or blog .... Benoit B. Mandelbrot. Benoit Mandelbrot is the Abraham Robinson Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus at the IBM T.J. Watson Research Center. ... us.macmillan.com/thefractalgeometryofnature - Cached - Similar 8. Fractals web hosting, domain name, free web site, email address ..... Benoit Mandelbrot is a physicist at the Thomas J. Watson Research Center of IBM, Yorktown Heights, New York, and a mathematician at Yale University. He originated Fractal ... www.fortunecity.com/emachines/e11/86/mandel.html - Cached - Similar 9. Finding Order in Chaos Fractal Geometry website at Yale University sponsored by Michael Frame, ... Benoit Mandelbrot's personal web page. Professor Mandelbrot is Sterling ... www3.wittenberg.edu/.../Chaos&FractalsHomePage.htm - Cached - Similar 10. NOVA: Hunting the Hidden Dimnesion: The self- similarity of Benoit ... Oct 28, 2008 ... That's the key to the self-similarity at the heart of fractals. ... Go to the companion Web site. NARRATOR: You can find it in the rain forest, ... BENOIT MANDELBROT (Yale University): I don't play with formulas, ... www.examiner.com/.../nova-hunting-the-hidden-dimnesion-the-self-similarity-of-benoit- mandelbrot-s-fractal-geometry - Cached 11. Books for benoit mandelbrot yale website fractal curriculum Fractals and Chaos: The Mandelbrot Set and Beyond - Benoit B Mandelbrot, CJG Evertsz, PW Jones, ... - 2004 - 326 pages The Fractal Geometry of Nature: Updated and ... - Benoit B Mandelbrot - 1982 - 500 pages Fractals Graphics and Mathematics Education - Michael Frame, Benoit B Mandelbrot, ... - 2002 - 232 pages

NOVA: Hunting the Hidden Dimnesion: The self-similarity of Benoit Mandelbrot's fractal geometry

 September 1st, 2009 5:35 pm PT Endless iterations of self-similar patters of vibrations: it's how we are becoming what we are

Benoit Mandelbrot opened to mathematics the wonderful world of wiggles.

My favorite philosopher, Alan Watts, often remarks on the regularity of human development, saying, "you can always tell when humans have been around, they're always trying to straighten things out." Perhaps fractal geometry holds a clue. Pay close attention to Keith Devlin: "think not of what you see, but what it took to produce what you see." That's the key to the self-similarity at the heart of fractals. Hunting the Hidden Dimension PBS Airdate: October 28, 2008 Go to the companion Web site

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.

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 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.

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 of Khan.

It was the first ever completely computer-generated 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 non-mathematical." What Mandelbrot said was that..."think not of what you see, but what it took to produce what you see."

NARRATOR: It takes endless repetition, and that gives rise to one of the defining characteristics of a fractal: what mathematicians call self-similarity.

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 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 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 human-made structures where we had straight lines and circles—and the perfect geometric shapes. The basic assumption that underlies classical mathematics is that everything is extremely regular. I mean, you reduce everything to straight lines...

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 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 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.