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Benoît Mandelbrot

October 18, 2010

My first office computer was a CP/M system in the era when no one had office computers. About a decade later, office computers were just coming into vogue, so everyone in our research and technology organization got an office computer. Surprisingly, these were Apple Macintosh computers, and it was suspected that this choice was made by a vice president who used a Mac. Years later, when the corporation tried to network us with its multitude of Windows computers and found it was too hard, we were forced to discard our Macs. I wasn't a Mac zealot, but the Macs worked quite well, and I developed quite a few Pascal utility programs on mine.

One thing I liked about my Mac was my screen saver. In those days, all displays were CRTs, and a static image would burn itself into the screen phosphor and be visible as a ghost image. The physics behind this is just the radiation damage from all the electrons. Burn-in is a real effect, as one of my laboratory terminals demonstrated. The monochrome terminal displayed an alphanumeric GUI I had programmed for an automation system, and most elements of this image were displayed continually. Not only was the ghost image of this control screen visible in the background of other programs, but it was visible even when the terminal was off. Forewarned, I bought the After Dark screen saver program. Yes, it had the flying toasters, but it had another animation that I really liked. This was called "Terraform." Terraform showed a moving strip of terrain, much like you would see from a satellite. These weren't displays of stored images. The terrains were generated using fractals. Benoît Mandelbrot, who coined the term, "fractal," and popularized the mathematics of fractals, died on October 14, 2010, at age 85.[1-8]

Benoît Mandelbrot

Benoît Mandelbrot


Mandelbrot was born in Warsaw to Lithuanian parents in 1924, just six days before my father. Like my father, who died in 2005, Mandelbrot died of cancer. Since his family was Jewish, they fled to France in 1936. Mandelbrot's early career was spent bouncing back and forth between France and the United States. After World War II, Mandelbrot studied at the École Polytechnique (Paris), and later earned a Master's degree in aeronautics from the California Institute of Technology. In 1952, he was awarded a Ph.D. in Mathematical Sciences from the University of Paris. Then it was a postdoctoral appointment under John von Neumann at the Institute for Advanced Study (Princeton, N.J.) where he likely got his first experience with computing machines.

After Princeton, Mandelbrot spent a few years at CNRS, the Centre National de la Recherche Scientifique, Paris, finally settling down at the IBM Thomas J. Watson Research Center (Yorktown Heights, N.Y) in 1957. He remained at IBM for many years, and his fractal work was done while he was at IBM. After retiring from IBM, he became a tenured professor at Yale University at age 75. His final position was the Sterling Professor Emeritus of Mathematical Sciences at Yale University.

Mandelbrot is best known for fractals and for popularizing ways to express the effective dimensionality of mathematical objects and Nature as numbers. Mandelbrot said that his interest in this topic arose from the idea of measuring the length of the coastline of Britain. A careful measurement on a wall map will give a number, but if you actually visit a section of the coast, you'll see that much fine structure was missed - an inlet here, an outcropping of rocks there. You find that the closer you examine the problem, the longer the coastline gets. A coastline is just one of many natural object with such a property. Mandelbrot used a cauliflower as another example.
"If you cut one of the florets of a cauliflower, you see the whole cauliflower but smaller. Then you cut again, again, again, and you still get small cauliflowers. So there are some shapes which have this peculiar property, where each part is like the whole, but smaller... So, the experience of humanity has always been that there are some shapes which have this peculiar property, that each part is like the whole, but smaller. Now, what did humanity do with that? Very, very little."[9]
Fractals, it was found, were important structures in science and economics. In economics, Mandelbrot, like his fellow Frenchman, Maurice Allais, was critical of the world's monetary system. He said that the mathematical models in use were insufficient to predict the complexity he discovered.

Mandelbrot did not practice mainstream mathematics. He decided to investigate areas with poorly defined problems that other mathematicians were loathe to pursue. His work with computers was something also that other mathematicians did not do. They championed the purity of proof and disdained what is now known as experimental mathematics. Let's admit it - Mandelbrot's 1982 book, "The Fractal Geometry of Nature," was a coffee table book, replete with pretty pictures held together by just a modicum of mathematics. However, his target audience wasn't mathematicians, who were generally dismissive of his work. Mandelbrot's career reminds me of both Paul Erdos and Carl Sagan. Erdos would discover many mathematical truths, but he would leave them unpublished. He felt no need to justify himself, and he thought it better to have a younger mathematician eventually publish the proof. Carl Sagan was a popularizer of science, and he was scorned by many of his peers for not spending his time on serious work. It's unbelievable, but Sagan was never elected to the National Academy of Sciences. Perhaps they were jealous of his fame.

Alfred Nobel apparently didn't think mathematics was that useful, so there's no Nobel Prize in mathematics. There's a story that Nobel didn't include mathematics as a prize category because a mathematician was having an affair with his wife. This is a nice story, but Nobel wasn't married. There are several prizes that second for a Nobel Prize in mathematics. These are the Abel Prize, the Fields Medal, and the Japan Prize. In 2003, Mandelbrot was awarded the Japan Prize, which is given to those who have "...advanced the frontiers of knowledge and served the cause of peace and prosperity for mankind." French President Nicolas Sarkozy said that France "is proud to have received Benoît Mandelbrot and to have allowed him to benefit from the best education."[7] For an entertaining snapshot of French education, click here for a short video.

There are quite a few fractal visualization programs available on the internet. One that runs nicely on my Ubuntu Linux system is GNU Xaos, a free program available for Windows, Mac OS X, Linux and BSD. If you ran fractal programs in the early days of computing, you'll be surprised at how quickly the present generation of personal computers renders these images. Here are two snapshots of the Mandelbrot Set rendered by GNU Xaos on my Ubuntu Linux PC.

Mandelbrot Set

A snapshot of the Mandelbrot Set.


Mandelbrot Set

A snapshot of the Mandelbrot Set.


References:

  1. Jascha Hoffman, "Benoît Mandelbrot, Mathematician, Dies at 85," New York Times, October 16, 2010.
  2. Edward Tenner, "Benoît Mandelbrot the Maverick, 1924-2010," The Atlantic, October 16, 2010
  3. Matt Blum, "He Gave Us Order Out of Chaos - R.I.P. Benoît Mandelbrot, 1924-2010," Wired, October 16, 2010.
  4. 'Fractal' mathematician Benoît Mandelbrot dies aged 85, BBC News, October 17, 2010.
  5. Mike James, "Benoît Mandelbrot, the father of Fractal Geometry, dies," i-Programmer, October 16, 2010.
  6. Michael Risinit, "Famous mathematician who once worked at IBM in Yorktown dies," Lower Hudson Journal, October 16, 2010.
  7. Mathematician Benoît Mandelbrot dies aged 85, Wikinews, October 17, 2010.
  8. Philippe Rivière, "La mort d'un mathématicien de génie - Benoît Mandelbrot change de dimension," Le Monde diplomatique, 16 octobre 2010.
  9. Benoît Mandelbrot: Fractals and the art of roughness, filmed February, 2010, at the Technology Entertainment and Design (TED) Conference.
  10. GNU Xaos Web Site.                                   

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Linked Keywords: CP/M; Apple Macintosh; Windows; Mac zealot; Pascal; screen saver; CRTs; screen_burn-in; phosphor; radiation damage; electrons; GUI; automation system; After Dark; flying toasters; fractals; Benoît Mandelbrot; Warsaw; Lithuanian; cancer; France; United States; World War II; École Polytechnique (Paris); Master's degree; aeronautics; California Institute of Technology; Ph.D.; Mathematical Sciences; University of Paris; John von Neumann; Institute for Advanced Study (Princeton, N.J.); Centre National de la Recherche Scientifique; IBM Thomas J. Watson Research Center (Yorktown Heights, N.Y); Yale University; Hausdorff dimension; measuring the length of the coastline of Britain; cauliflower; economics; Maurice Allais; proof; experimental mathematics; The Fractal Geometry of Nature; coffee table book; Paul Erdos; Carl Sagan; National Academy of Sciences; Alfred Nobel; Abel Prize; Fields Medal; Japan Prize; Nicolas Sarkozy; GNU Xaos; Mandelbrot Set.

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