3. The Mandelbrot Set and Julia Sets

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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 Dynamical Systems and Technology Project website.

Discovered around 1980, the Mandelbrot set may well be the most familiar image produced by the mathematics of the last century. Its status as a cultural icon needs no support.

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.

Also, 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.

A. 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. B. 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. C. The Mandelbrot set. 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. D. Combinatorics of the Mandelbrot Set. Associated with each disc and cardioid of the Mandelbrot set is a cycle. There are simple rules relating the cycle of a feature to those of nearby features. From this we can build a map of the Mandelbrot set. E. Some features of the Mandelbrot set boundary. The boundary of the Mandelbrot set contains infinitely many copies of the Mandelbrot set. In fact, as close as you look to any boundary point, you will find infinitely many little Mandelbrots. The boundary is so "fuzzy" that it is 2-dimensional. Also, the boundary is filled with points where a little bit of the Mandelbrot set looks like a little bit of the Julia set at that point. F. Scalings in the Mandelbrot Set. The Mandelbrot set includes infinitely many smaller copies of itself. These can be organized into hierarchical sequences for which the ratio of the sizes of successive copies approaches a limiting value. Some of these give the Feigenbaum constant associated with the logistic map, others give new constants. Some give integer limits. G. Complex Newton's Method. Julia sets related to finding the roots of equations. SImilar features arise in magnetic pendula and in light reflected within a pyramid of shiny spheres. H. Universality of the Mandelbrot Set. Newton's method for a family of cubic polynomials revealed more copies of the Mandelbrot set. Yet Newton's method is nothing like z → z2 + c. Further investigation shows we're surrounded by Mandelbrot sets. I. Here is Ray Girvan's page on the Mandelbrot Monk. Was the Mandelbrot set discovered in the 13th century? Read the page carefully. J. Fractals in Literature. Not only are fractals present in the structure of literature, sometimes they are the subject of literature. K. Fractals in Art. Because it exhibits a balance of familiarity and novelty, the Mandelbrot set is more interesting than the Sierpinski gasket. This aesthetic maxim is familiar in art. Here is a recent class project on the Mandelbrot set.