The Most Famous Fractal
Largely because of its haunting beauty, the Mandelbrot set has become the most famous object in modern mathematics. It is also the breeding ground for the world's most famous fractals. Since 1980, the set has provided an inspiration for artists, a source of wonder for schoolchildren, and a fertile testing ground for the science of linear dynamics.
The set itself is a mathematical artifact—an odd-shaped infinite swarm of points clustered on what is known as the "complex number plane." Let's try to visualize it.
Picturing the plane
To make them tangible, we imagine real numbers like 1, 2, 3... as spaced out along a number line. Because complex numbers have two parts to them—called their "real" and "imaginary" parts—making complex numbers tangible requires two lines, or axes, which means a plane. Picture the plane dotted by complex numbers as a computer screen, which is just where the visual form of the Mandelbrot set was discovered.
Like the screen of your television set, a computer screen is covered with a host of very tiny, evenly spaced points, called pixels. The moving image on the screen is made when patterns of pixels are excited (made to glow) by a fast-moving scanning beam of electrons. Think of each pixel as a complex number. The pixels in any neighborhood are numerically close to each other, just as 3 and 4 are numerically close to each other on the real number line. Pixels (numbers) are made to glow by applying an iterative equation to them.
In the late 1970s and early 1980s Benoit Mandelbrot, the inventor of fractal geometry, and several others were using simple iterative equations to explore the behavior of numbers on the complex plane. [Read an interview with Mandelbrot.] A very simple way to view the operation of an iterative equation is as follows:
Start with one of the numbers on the complex plane and put its value in the "Fixed Number" slot of the equation. In the "Changing Number" slot put zero. Now calculate the equation, take the "Result," and slip it into the "Changing Number" slot. Repeat the whole operation again (in other words, recalculate and "iterate" the equation) and watch what happens to the "Result." Does it hover around a fixed value, does it spiral toward infinity quickly, or does it stagger upward by a slower expansion?
A mathematical marvel
When iterative equations are applied to points in a certain region of the complex plane, the results are spectacular. By treating the pixels on computer screens as points on the plane, even nonmathematicians can now admire this marvel. In fact, without computers, only the most intuitive mathematicians could have glimpsed what was there. With the computer it works like this:
Starting with the value of a point (or pixel) and applying the equation to it, iterate the equation perhaps 1,000 times. If the "Result" remains stable, color the pixel black. If the number heads at one speed or another toward infinity, paint it a different color, assigning colors for each rate of movement. The points (pixels) representing the fastest-expanding numbers might be colored red, slightly slower ones magenta, very slow ones blue—whatever color scheme the fractal explorer decides. Now move on to the next pixel and do the same thing with the color palette until all the pixels on the screen have been colored.
Artists and the public have been attracted by the set's haunting beauty.
When all the pixels (or points representing complex numbers) have been iterated by the equation, a pattern emerges. The pattern that Mandelbrot and others discovered in one region of the complex plane was a long-proboscidean insect shape of stable points—the Mandelbrot set itself, usually shown in black—surrounded by a flaming boundary of filigreed detail that includes miniature, slightly distorted replicas of the insect shape, and layer upon layer of self-similar forms.
The boundary area of the set is infinitely complex, therefore fractal, because it is possible to bring out finer and finer detail. Computer graphics artists call the process of unfolding the detail "zooming in" on the set's boundary or "magnifying" it. It's fairly easy to grasp what this means.
On the real-number line we routinely imagine that between the numbers 1 and 2 are other numbers, 1.5, for example, or 1.6. (We encounter this every time we pick up a ruler.) Of course, between those numbers are still more numbers—1.53 and 1.54, for example—and so on, indefinitely. The same is true for the numbers on the complex plane. Between any two of them are many more, and between those many more are many more still, ad infinitum. These numbers between numbers allow us to use the computer like a microscope to dive into increasingly deeper detail. [Dive in yourself with A Sense of Scale.]
To extend our analogy, if the numbers we were examining on the complex plane were all like the numbers at the level of, say, 1, 2, 3, etc., on a ruler, then we would be examining the largest scale of numbers. But we could also go to a smaller scale and examine the numbers at the level of 1.5, 1.6. Between those will be yet a smaller scale (including the numbers 1.53 and 1.54, for example), and so in any region of the complex plane we could move downward (or inward) to smaller and smaller scales.
Similarly, explorers of the Mandelbrot set can zoom in to study finer and finer detail as they examine the ever smaller scales of numbers between numbers on the complex plane. Indeed, a home computer can examine numbers out to 15 decimal points. To complete the microscope analogy, if the numbers 1 and 2 were the equivalent of objects the size of human beings and trees, a number 15 decimal points smaller would be an object tinier than an atom. More powerful computers can go into even finer (or deeper) detail. In addition, different styles of iterative equations can act as prisms to display varying facets of the behavior of the complex numbers around the set.
Applying zoom-ins and different iterative prisms to the numbers in the boundary area of the Mandelbrot set has revealed that this region is a mathematical strange attractor. The "strange attractor" name here applies to the set because it is self-similar at many scales, is infinitely detailed, and attracts points (numbers) to certain recurrent behavior. Scientists study the set for insights into the nonlinear (chaotic) dynamics of real systems. For example, the wildly different behavior exhibited when two numbers with almost the same starting value and lying next to each other in the set's boundary are iterated is similar to the behavior of systems like the weather undergoing dynamic flux because of its "sensitive dependence on initial conditions."
But a major importance of the set may be that it has become a strange attractor for scientists, artists, and the public, though each may be drawn to it for quite different reasons. Scientists have found themselves attracted—often with childlike delight—to a new aesthetic that involves the artistic choices of color and detail they must make when exploring the set. Artists and the public have been attracted by the set's haunting beauty and the idea of abstract mathematics turned into tangible pleasures. [Make your own tangible pleasures—see Design a Fractal.]
Homer Smith, cofounder of an independent research group based at Cornell University, produces fractal images with the aim of attracting young children to mathematics. [See some of Smith's stunning images in A Radical Mind.] "We hope that fractals show up in early classrooms, to get kids interested in mathematics very early," Smith says, "because it really opens the eyes of children who haven't been turned off by education... We hope by the time they get up to the tenth grade, they'll have seen these things and say, 'There's something here in math, science, and computers that I want to learn.'"
In this detail of the Mandelbrot set, the set itself appears in black, with the fractal boundary alive with color.
Because an infinite number of points exist between any two points on the number plane, the Mandelbrot set's detail is infinite. This image is a tiny part of the previous image magnified many thousands of times over.
"A flaming boundary of filigreed detail" is how Briggs aptly describes the border of the Mandelbrot set.
The "self-similar" nature of fractals means that particular elements, such as the Mandelbrot set, reappear over and over again, no matter how "deep" one goes into the image through magnification.
John Briggs is author of Fractals: The Patterns of Chaos (Simon & Schuster, 1992), from which this article was excerpted with kind permission of the author and publisher.
© | Created October 2008