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Friday, June 02, 2006
Nonlinearity
PSAYINGS.5A.41. The standard graphic representation of the Mandelbrot set is really only a snapshot of what it entails. The set is infinite and has the interesting property of painting the same shapes, including the one shown above, no matter how small a sample of it you take. It just keeps going. The history of the set used to be pretty straightforward:
Discovered in 1976 by IBM researcher
Benoit Mandelbrot, the Mandelbrot set is the most famous fractal (a
mathematical object with the property of infinite detail). Only the
advent of fast computers made feasible the repeated calculations
involved -- or so it was thought.
The problem is, a 13th century monk named Udo seems to have anticipated
Mandelbrot by about 700 years, if mathematician Robert Schipke is
to be believed. He found the set standing in for the Star of Bethlehem
in a drawing of the nativity by Udo of Aachen. Udo was already known to
moderns as the author of the poem Carl Orff set to music in the Carmina Burana. What no one knew
was that the stirring O Fortuna
of the opera was as much literal and mathematical as it was poetic: Udo
had spent much of his youth exploring probability theory in the hope of
learning how to predict which souls would go to heaven and which to
hell. This led him to devise the exact equation that produces the
Mandelbrot set. He spent years performing calculations by hand that the
20th century IBM mathematician would later complete in a matter of
hours on a computer. His patience was rewarded with a vision of the
intriguing shape that has played across computer screens for the past 30
years.You can read the rest of Udo's story at the website linked above. We found it fascinating because the Mandelbrot set is an outgrowth of Chaos theory, which is responsible for much of the structure of The Boomer Bible, as you'll see here. Another name for chaos is nonlinear systems theory, because it deals with the messy reality of equations that don't come out even, and where the finite can suddenly veer off the chart into infinity.
Which is kind of what happened with Udo, as you'll learn here.
All in all, too good not to share.
