Tuesday, Nov 27, 2012

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These images come from the Mandelbrot set gallery of Frank Roussel,

http://graffiti.u-bordeaux.fr/MAPBX/roussel/fractals.html.

Applets to explore Julia sets and the Mandelbrot set, and other fractal topics, can be found at Bob Devaney's website

Dynamical Systems and Technology Project

We have our own piece of Mandelbrot set software.

From a philosophical perspective, the Mandelbrot set challenges familiar notions of simplicity and complexity: how could such a simple formula, involving only multiplication and addition, produce a shape of great organic beauty and infinite subtle variation?

Also, deep mathematics underlies the Mandelbrot set. Despite years of study by brilliant mathematicians (three of whom won Fields Medals), some natural and simple-to-state questions remain unanswered. Much of the rebirth of interest in complex dynamics was motivated by efforts to understand the stunning images of the Mandelbrot set.

In addition, as we shall see, hidden within it are metaphors (and more) for some of the richness of contemporary literature and music.

Finally, some instances are just plain entertaining, in one way or another.

Complex Iteration. A review of complex arithmetic: the background needed to use the formulas that generate pictures of Julia sets and of the Mandelbrot set.

Julia Sets. For a complex number c, the filled-in Julia set of c is the set of all z for which the iteration z → z² + c does not diverge to infinity. The Julia set is the boundary of the filled-in Julia set. For almost all c, these sets are fractals.

The Mandelbrot set is the set of all c for which the iteration z → z² + c, starting from z = 0, does not diverge to infinity. Julia sets are either connected (one piece) or a dust of infinitely many points. The Mandelbrot set is those c for which the Julia set is connected. Here is a careful definition, a gallery of images, and claim of an early discovery of the Mandelbrot set. For this last, check the posting date of the source.

Complex Iteration. A review of complex arithmetic: the background needed to use the formulas that generate pictures of Julia sets and of the Mandelbrot set.
Julia Sets. For a complex number c, the filled-in Julia set of c is the set of all z for which the iteration z → z² + c does not diverge to infinity. The Julia set is the boundary of the filled-in Julia set. For almost all c, these sets are fractals.
The Mandelbrot set is the set of all c for which the iteration z → z² + c, starting from z = 0, does not diverge to infinity. Julia sets are either connected (one piece) or a dust of infinitely many points. The Mandelbrot set is those c for which the Julia set is connected. Here is a careful definition, a gallery of images, and claim of an early discovery of the Mandelbrot set. For this last, check the posting date of the source.