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I’LL BET you never thought about fitting an infinite length onto something the size of a postage stamp. However, this is today’s topic, thanks to Benoit B. Mandelbrot, the Polish-born French and American mathematician who discovered fractals. It happens that his posthumous memoir, The Fractalist: Memoir of a Scientific Maverick, is reviewed in Science magazine, 18 January 2013, published by the American Association for the Advancement of Science.
Mandelbrot called himself a maverick because, early on, he sensed his life’s best achievements would be on the ragged fringe of academe, not hemmed in by any particular discipline.
Working at IBM’s Thomas J. Watson Research Center for 35 years, Mandelbrot enriched areas as varied as geology, medicine, chaos theory, fluid dynamics, information theory and economics.
As a Visiting Professor at Harvard (in Economics), he applied his computer-graphic expertise in studying bizarre geometric transformations of shapes, mathematical objects now known as fractals. Mandelbrot coined the term in 1975, based on its similarity to the word “fractured.”
Think of a fractal as an infinitely self-repeating shape. Imagine a sort of Google Map view that gets increasingly magnified—even down to the molecular level.
Many fractals involve deep mathematics. However, one known as the Koch Snowflake is quite straightforward. Discovered by Swedish mathematician Helge von Koch in 1904, his Snowflake provides an example of an infinite length on a postage stamp.
The Koch Snowflake begins with a triangle of three equal sides. Now erase the middle third of each side of this equilateral triangle and, on the open space, erect an equilateral shape of smaller size. Thus far, the result is a six-pointed star.
Next, repeat the process on each segment of this figure: Erase the middle thirds and erect (twelve) even smaller shapes.
And then do it again, and again, and again. The result is a Koch Snowflake.
To prove my claim about infinite length, let’s set the initial perimeter at 1 unit. Then it’s not difficult to see the new perimeter loses some length but adds more, specifically 1 – 3/9 + 6/9 = 1 – 1/3 + 2/3 = 1 + 1/3 = 4/3.
It’s only some tedious fractional arithmetic away from showing that, at the nth iteration, the perimeter is (4/3)n.
Last, it’s easy to recognize that as n gets larger, so does (4/3)n. Technically, as n → ∞, then (4/3)n → ∞ as well.
Thus, start with an equilateral triangle drawn on a postage stamp, and its related Koch Snowflake stays within the stamp—yet has infinite length.
Neat, eh? And only one example of the beauty of fractals.
Mandelbrot discovered a rather more elaborate fractal, named the Mandelbrot Set in his honor, in 1979. He was 55 at the time. A rarity among mathematicians, he accomplished some of his finest work in middle age and beyond. Mandelbrot finally made tenure in 1999, in the Department of Mathematics, Yale University—at age 75.
Because of his many varied achievements in mathematics—and outside this realm—Mandelbrot is considered one of the most important thinkers of the 20th century. ds
© Dennis Simanaitis, SimanaitisSays.com, 2013