HONR 300/CMSC 491 Computation, Complexity, and Emergence Mandelbrot & Julia Sets Prof. Marie desJardins February 22, 2012 Based on slides prepared by Nathaniel Wise

Chapter 8: The Mandelbrot Set & Julia Sets There once was a young man from Trinity Who took. But the number of digits Gave him the fidgets; He dropped Math and took up Divinity.

A New Kind of Fractal The fractals we've looked at are generally self-identical, in that you can look at them at different scales and they look exactly the same. The Mandelbrot and Julia sets are only self-similar: they have a kind of pattern that's instantly recognizable, at every scale, but no two scales are exactly the same. We'll see this kind of “different sameness” again when we start to look at chaos theory and chaotic systems.

The Mandelbrot Set Benoit Mandelbrot ( ) is known as the “father of fractal geometry.” He invented the term “fractal,” and used the new field of computation and digital computers to explore complex mathematical objects that had previously only been studied in the abstract. The Mandelbrot set is defined using an iterative function: x t+1 = x t + c, where x t = 0. The magnitude of a complex number a + bi, is the Euclidean distance of that point from the origin of the complex plane, i.e., √a 2 + b 2 For a given value c, it turns out that the magnitude of x t+1 will do one of two things: It will always be smaller than 2 (no matter how large t gets), or It will eventually diverge (i.e., x t will go to ∞ as t goes to ∞). The Mandelbrot set is defined as the set of values c for which x t+1 remains smaller than 2.

Computing the Mandelbrot Set The Mandelbrot set contains those values of c for which the magnitude x t remains smaller than 2 for all t. But we have no easy way to know whether the Mandelbrot series diverges for a given value of c! If we compute the Mandelbrot series for some value c and the magnitude of x t ever becomes greater than 2, that value c is definitely not in the Mandelbrot set. (It is a property of the series that if x t is greater than 2, then subsequent values will always increase.) But a Mandelbrot series may remain below 2 for arbitrarily long before diverging, and the only way to tell if it will diverge is to compute the sequence for long enough.

The black area corresponds to points in the Mandelbrot set. The colored area represents points not in the Mandelbrot set, where the brightness of the color is proportional to the number of iterations before divergence (i.e., the smallest value of t for which x t ≥ 2).

Julia Sets Long before Mandelbrot, Gaston Julia ( ) had studied a similar function. (In fact, Mandelbrot started out by studying the Julia set...) Here, c is fixed complex number (so we talk about “the Julia set for c = some value”) and x 1 is the point being examined (i.e., the point that is plotted in a display of the Julia set as belonging to that Julia set (or not)). Julia examined what happens to the series for a given c and x 1 as i increases. As with points in the Mandelbrot set, each such series either diverges, or it does not. Without the aid of computers, Julia could only sketch relatively crude drawings of these shapes. Today, we can compute the Julia set for any value, to an arbitrary degree of resolution.

Julia Sets c = ic = i The central black areas are points that converge and are a part of the set. The different colors represent how many iterations before that point diverges.

Julia Sets c = ic = i The central black areas are points that converge and are a part of the set. The different colors represent how many iterations before that point diverges.

Julia Sets c = ic = i The central black areas are points that converge and are a part of the set. The different colors represent how many iterations before that point diverges.

The Mandelbrot Set Some Julia sets consist of infinitely many disconnected regions; others are a single contiguous region (although they may be connected only by arbitrarily fine “filaments”). The Mandelbrot set serves as a “map” of all the Julia sets. If a point is inside the Mandelbrot set (colored black), then the corresponding Julia set is contiguous. The closer a point is to any border area of the Mandelbrot set, the more complex that Julia set will be. Julia sets often seem to share similar visual characteristics to the corresponding point in the Mandelbrot set. The NetLogo model posted on the course page lets you explore the Mandelbrot set and corresponding Julia sets:

The Mandelbrot Set The Mandelbrot set is perhaps the most complex object in mathematics. One could spend a lifetime exploring it and never see all of it. It contains infinitely many imperfect copies of the set within it, none of them matching any other copy. YouTube user ckorda spent 5 months with about 15 PCs all rendering a video of a Mandelbrot zoom to a depth of (about ): One can zoom as far as your computing power and patience holds up: the NetLogo model can do up to a one-billion zoom, depending on the region.