Tuesday, October 19, 2010

Fractals: Mandelbrot

Fractals And Foodspotting
FoodSpotting Is The Next FourSquare
Fractals: Apple, Windows 95, Netscape, Google, Facebook, Twitter

PBS: Goodbye to the Father of Fractals: "A mind that is stretched by a new experience can never go back to its old dimensions." .... the mathematician Benoit Mandelbrot, who died last week at the age of 85. .... Mandelbrot invented the term fractal to describe the "roughness" he saw all around him in nature -- the jagged shape of a cloud, the rugged indentations of a coastline. Classical Euclidean mathematics, the kind we learn in school, serves well for the human-made world of straight lines, circles, and squares. But nature's non-linear shapes were generally considered unmeasurable -- until Mandelbrot developed fractal geometry. ..... "In the whole of science, the whole of mathematics, smoothness was everything," Mandelbrot says in "Hunting the Hidden Dimension." "What I did was open up roughness for investigation ...... required rethinking one of the basic concepts in
Image representing Foodspotting as depicted in...Image via CrunchBase math -- dimension. ..... another dimension exists between two and three dimensions. It's a fractal dimension, and the rougher something is, the higher its fractal dimension. This roughness, this fractal dimension, he discovered, could be measured quite well using fractal geometry. ..... they're just at different scales. They're self-similar. .... his classic 1982 book The Fractal Geometry of Nature .... From the self-similar stalks on a head of broccoli, to the ever-smaller branching of blood vessels in the human body, fractals are everywhere in nature. ..... his new math has informed fields as diverse as biology and physics, ecology and engineering, medicine and cosmology. ...... the end of its applicability is nowhere in sight.
A Radical Mind: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line." .... a new kind of visual mathematics ..... Fractal geometry, as he called this new math, is worlds apart from the Euclidean variety we all learn in school, and it has sparked discoveries in myriad fields, from finance to metallurgy, cosmology to medicine. ...... why he abandons work on any given advance in fractals as soon as it becomes popular. ..... "I heard fractals described endless times as 'pretty pictures but pretty useless,'" recalls Benoit Mandelbrot, the inventor of fractal geometry. Mandelbrot knew better, and today applications for fractals are, he says, "so numerous and diverse you wouldn't believe it." ...... all kinds of phenomena in which irregularity and variability dominate but are so great that they don't average out. ...... Galileo's famous saying that the Great Book of Nature is written in the language of mathematics ...... the great bulk of science studies smooth behavior, in particular using equations that assume that everything evolves in a very regular fashion. ...... I greatly learned from both the kind and the unkind comments I continually received. ..... But I was so alone that the direction I was following was not described by any existing word. In 1975, my work forced me to coin one: fractal. ..... "With two hands, you can count all the simple shapes of nature. Everything else is rough." ..... a host of phenomena that for me smelled the same ..... my study of fractals began with the stock market ..... Fractal antennas are now almost routine. .... Concrete was known to the Romans, then was forgotten and had to be reinvented. ...... an altogether new kind of concrete that is enormously stronger and more durable, a form entirely based upon his deep understanding of fractals. ...... a wall having a fractal surface would be far b

Image representing Foursquare Solutions as dep...Image via CrunchBaseetter because it would absorb the noise. ..... if you look around us, almost everything industrial is very smooth, round, flat, corrugated, and so on. Now that is changing. Engineers everywhere know how to use fractals. ...... how uneven a healthy heartbeat is. ..... Read everything written about fractals? I don't even try. ...... I abandon problems when a constituency gets created around them. ...... I can stand loneliness. In fact, I'm rarely comfortable in a big crowd, because big crowds automatically are very specifically organized by dates, by tradition, by training. And I don't sound like a mathematician. I don't sound like a physicist either. Nor do I sound like an art critic. There's very great strength in being a stranger, if one brings something new. ........ a maverick. .... I was at least twice a foreigner ..... We didn't read newspapers by curiosity but because, within a week or a month, what we read could affect our whole life. ...... We moved to France when my mother, at age 50, made a decision I still find admirable and incomprehensible. She abandoned her profession, her roots, her friends, and her place in a well-defined society to become a lonely housewife in a slum in Paris. My parents figured this gave us a better chance to survive. ...... A constant feeling of anxiety made me develop what many people call a survival instinct. ..... In France, one must belong to one well-defined and very stable sub-community. ..... France had no counterpart to James Maxwell in Britain or Max Planck in Germany, who were creating theoretical physics. ...... my early rootlessness gave me an awareness that one can live without being so completely specified. ..... Newton was called a natural philosopher. And in the 18th century, the professions of mathematics and physics were not deeply distinguished ..... I have always been extremely disciplined and conservative. ...... "What I did was totally despised by my peers, who felt that I'd completely destroyed my promise." ..... One very close friend, whom I admired endlessly and rated as superior to me in many of the basic tools of our trade, never knew how to discipline himself. By the end of his life, he had already been forgotten. ..... in the mood that prevailed in France, I belonged nowhere ..... A good older friend commented that my Ph.D. was one half in a field that didn't yet exist, and one half in a field that no longer existed. ....... an intense feeling of not belonging. .... so I thought I might, perhaps for a couple of years, do enough to please the establishment, yet save enough time to do some gambles of mine. ....... "It dawned on him that I was unpredictable and might do things about which he didn't give a hoot." ....... everywhere else I had this problem of being unclassifiable ....... People would tell me, "If we knew that you were going to stop running around and, from today on, become an honest economist or an honest electrical engineer or an honest this or that, we would offer you a job instantly." ....... a proposal I made to the National Science Foundation was turned down. I was very surprised when a man telephoned and told me, "In order to fund this thing, we need six Outstanding reviews, and you have five Outstanding and one Excellent. So you can't be funded." ....... In economics there are no proofs. Science is "proven" by its applicability. ...... But they don't have the creativity to ask new questions. ..... The greatest mathematician in my private pantheon has been Henri Poincaré. ..... he started many branches of mathematics from scratch ..... he didn't prove any difficult theorem and cared about proofs less than about concepts

Wikipedia: Mandelbrot Set
Wikipedia: Benoît Mandelbrot

Chaos Theory: A Brief Introduction | IMHO: The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)
Video: Web of Stories • Benoît Mandelbrot • The butterfly effect
RIP Benoît Mandelbrot, Genius of Roughness, Connoisseur of Chaos
Mandelbrot Set Algorithm - Butterfly Effect
Chaos theory - Wikipedia, the free encyclopedia
Scientific American: Tribue To Mandelbrot « The Swapper
EYE ON MIAMI: Fractals, Chaos Theory and the Butterfly Effect

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