Benoit B. Mandelbrot (1924-2010)

摘要

Mandelbrot coined the term "fractals" to describe geometrical objects that cannot be simplified by magnification. A fractal is, loosely, a subset of a real space that has non-integer Hausdorff-Besicovitch dimension. Some such sets, including the Cantor set, the Sierpinski triangle, and Julia sets, were known to mathematics, but did not see the light of day until the early 1980s, aided by computer graphics. A famous fractal is the Mandelbrot set, endless zooms of which can be readily viewed on YouTube. The Mandelbrot set connects fractal geometry to the chaotic behaviour of quadratic dynamical systems: it is a map of the locus of connectivity of the repelling set for the complex Logistic map, and has been the subject of much deep mathematical research. Such objects relate to Leibnitz's attempt to tighten Euclid's axioms: "the straight line is a curve any part of which is similar to the whole, and it alone has this property, not only among curves but among sets". In practice, fractals are complicated geometrical objects that can be produced by random or deterministic iteration of simple formulas. They are used to model diverse observable phenomena.