Figure 6-12
Geometric operation for constructing a Koch curve.
With the help of computers, simple geometric iterations can be applied thousands of times at different scales to produce so-called fractal forgeries—computer-generated models of plants, trees, mountains, coastlines, and so on that bear an astonishing resem-
blance to the actual shapes found in nature. Figure 6-15 shows an example of such a fractal forgery. By iterating a simple stick drawing at various scales, the beautiful and complex picture of a fern is generated.
Figure 6-15
Fractal forgery of a fern; from Garcia (1991).
With these new mathematical techniques scientists have been able to construct accurate models of a wide variety of irregular
natural shapes and in so doing have discovered the pervasive appearance of fractals. Of all those, the fractal patterns of clouds, which originally inspired Mandelbrot to search for a new mathematical language, are perhaps the most stunning. Their self-similarity stretches over seven orders of magnitude, which means that the border of a cloud magnified ten million times still shows the same familiar shape.
Complex Numbers
The culmination of fractal geometry has been Mandelbrot’s discovery of a mathematical structure that is of awesome complexity and yet can be generated with a very simple iterative procedure. To understand this amazing fractal figure, known as the Mandelbrot set, we need to first familiarize ourselves with one of the most important mathematical concepts—complex numbers.
The discovery of complex numbers is a fascinating chapter in the history of mathematics. 28 When algebra was developed in the Middle Ages and mathematicians explored all kinds of equations and classified their solutions, they soon came across problems that had no solution in terms of the set of numbers known to them. In particular, equations like x + 5 = 3 led them to extend the number concept to negative numbers, so that the solution could be written as x = —2. Later on, all so-called real numbers—positive and negative integers, fractions and irrational numbers (like square roots, or the famous number tt)— were represented as points on a single, densely populated number line (figure 6-16).
- 5/2
V2 V?
77
0
With this expanded concept of numbers, all algebraic equations could be solved in principle except for those involving square roots
THE MATHEMATICS OF COMPLEXITY 143
of negative numbers. The equation x 2 = 4 has two solutions, x = 2 and x — —2, but for x 2 — —4 there seems to be no solution, because neither +2 nor -2 will give -4 when squared.
The early Indian and Arabic algebraists repeatedly encountered these equations, but they refused to write down expressions like V — 4 because they thought them to be completely meaningless. It was not until the sixteenth century that square roots of negative numbers appeared in algebraic texts, and even then the authors
were quick to point out that such expressions did not really mean anything.
Descartes called the square root of a negative number “imagi- nary and believed that the occurrence of such 4 imaginary” numbers in a calculation meant that the problem had no solution. Other mathematicians used terms such as “fictitious,” “sophisticated,” or “impossible” to label those quantities that today, following Descartes, we still call “imaginary numbers.”
Since the square root of a negative number cannot be placed anywhere on the number line, mathematicians up to the nineteenth century could not ascribe any sense of reality to those quantities. The great Leibniz, inventor of the differential calculus, attributed a mystical quality to the square root of—1, seeing it as a manifestation of “the Divine Spirit” and calling it “that amphibian between being and not-being.” 29 A century later Leonhard Euler, the most prolific mathematician of all time, expressed the same sentiment in his Algebra in words that, even though less poetic, still echo the same sense of wonder:
All such expressions as 1, /~^2, etc., are consequently impossible, or imaginary numbers, since they represent roots of negative quantities; and of such numbers we may truly assert that they are neither nothing, nor greater than nothing, nor less than nothing, which necessarily constitutes them imaginary or impossible. 30
In the nineteenth century another mathematical giant, Karl Friedrich Gauss, finally declared forcefully that “an objective existence can be assigned to these imaginary beings.” 31 Gauss real-
ized, of course, that there was no room for imaginary numbers anywhere on the number line, so he took the bold step of placing them on a perpendicular axis through the point zero, thus creating a Cartesian coordinate system. In this system all real numbers are placed on the “real axis” and all imaginary numbers on the “imaginary axis” (figure 6-17). The square root of — 1 is called the “imaginary unit” and given the symbol i, and since any square root of a negative number can always be written as J —a — /—I J~a = if a, all imaginary numbers can be placed on the imaginary axis as multiples of z.
Figure 6-17 The complex plane.
With this ingenious device Gauss created a home not only for imaginary numbers, but also for all possible combinations of real and imaginary numbers, such as (2 + z), (3 - 2z), and so on. Such combinations are called “complex numbers” and are represented
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THE MATHEMATICS OF COMPLEXITY
by points in the plane spanned by the real and imaginary axes, which is called the “complex plane.” In general, any complex number can be written as
z = x + iy
where x is called the real part” and y the “imaginary part.”
With the help of this definition Gauss created a special algebra of complex numbers and developed many fundamental ideas about functions of complex variables. Eventually this led to a whole new branch of mathematics, known as “complex analysis,”
which has an enormous range of applications in all fields of science.
Patterns within Patterns
The reason why we took this excursion into the history of complex numbers is that many fractal shapes can be generated mathematically by iterative procedures in the complex plane. In the late seventies, after publishing his pioneering book, Mandelbrot turned his attention to a particular class of those mathematical fractals known as Julia sets. 32 They had been discovered by the French mathematician Gaston Julia during the early part of the century but had soon faded into obscurity. In fact, Mandelbrot had come across Julia s work as a student, had looked at his primitive drawings (done at that time without the help of a computer), and had soon lost interest. Now, however, Mandelbrot realized that Julia’s drawings were rough renderings of complex fractal shapes, and he proceeded to reproduce them in fine detail with the most powerful computers he could find. The results were stunning.
The basis of the Julia set is the simple mapping
z —> z 2 + c
where z is a complex variable and c a complex constant. The iterative procedure consists in picking any number z in the complex plane, squaring it, adding the constant c, squaring the result again, adding the constant c once more, and so on. When this is
done with different starting values for z, some of them will keep increasing and move to infinity as the iteration proceeds, while others will remain finite. 33 The Julia set is the set of all those values of z, or points in the complex plane, that remain finite under the iteration.
To determine the shape of the Julia set for a particular constant c, the iteration has to be carried out for thousands of points, each time until it becomes clear whether they will keep increasing or remain finite. If those points that remain finite are colored black, while those that keep increasing remain white, the Julia set will emerge as a black shape in the end. The entire procedure is very simple but very time-consuming. It is evident that the use of a high-speed computer is essential if one wants to obtain a precise shape in a reasonable time.
Figure 6-18
Varieties of Julia sets; from Peitgen and Richter (1986).
For each constant c one will obtain a different Julia set, so there is an infinite number of these sets. Some are single connected pieces; others are broken into several disconnected parts; yet others look as though they have burst into dust (figure 6-18). All have the
jagged look that is characteristic of fractals, and most of them are impossible to describe in the language of classical geometry. “You obtain an incredible variety of Julia sets,” marvels French mathematician Adrien Douady. “Some are a fatty cloud, others are a skinny bush of brambles, some look like the sparks which float in the air after a firework has gone off. One has the shape of a rabbit, lots of them have seahorse tails.” 34
This rich variety of forms, many of which are reminiscent of living things, is amazing enough. But the real magic begins when we magnify the contour of any portion of a Julia set. As in the case of a cloud or coastline, the same richness is displayed across all scales. With increasing resolution (that is, with more and more decimals of the number z entering into the calculation) more and more details of the fractal contour appear, revealing a fantastic sequence of patterns within patterns—all similar without ever being identical.
When Mandelbrot analyzed different mathematical representations of Julia sets in the late seventies and tried to classify their immense variety, he discovered a very simple way of creating a single image in the complex plane that would serve as a catalog of all possible Julia sets. That image, which has since become the principal visual symbol of the new mathematics of complexity, is the Mandelbrot set (figure 6-19). It is simply the collection of all points of the constant c in the complex plane for which the corresponding Julia sets are single connected pieces. To construct the Mandelbrot set, therefore, one needs to construct a separate Julia set for each point c in the complex plane and determine whether that particular Julia set is “connected” or “disconnected.” For example, among the Julia sets shown in figure 6-18, the three sets in the top row and the one in the center panel of the bottom row are connected (that is, they consist of a single piece), while the two sets in the side panels of the bottom row are disconnected (consist of several pieces).
To generate Julia sets for thousands of values of c, each involving thousands of points requiring repeated iterations, seems an impossible task. Fortunately, however, there is a powerful theo-
Figure 6-19
The Mandelbrot set; from Peitgen and Richter (1986).
rem, discovered by Gaston Julia himself, which drastically reduces the number of necessary steps. 35 To find out whether a particular Julia set is connected or disconnected, all one has to do is iterate the starting point z = 0. If that point remains finite under repeated iteration, the Julia set is always connected, however crumpled it may be; if not, it is always disconnected. Therefore one really needs to iterate only that one point, z = 0, for each value of c to construct the Mandelbrot set. In other words, generating the Mandelbrot set involves the same number of steps as generating a Julia set.
While there is an infinite number of Julia sets, the Mandelbrot set is unique. This strange figure is the most complex mathematical object ever invented. Although the rules for its construction are very simple, the variety and complexity it reveals upon close inspection is unbelievable. When the Mandelbrot set is generated on a rough grid, two disks appear on the computer screen: the smaller one approximately circular, the larger one vaguely heart shaped. Each of the two disks shows several smaller disklike at-
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tachments to its boundary, and further resolution reveals a profusion of smaller and smaller attachments looking not unlike prickly thorns.
Figure 6-20
Stages of a journey into the Mandelbrot set. In each picture the area of the Subsequent magnification is marked with a white rectangle; from Peitgen and Richter (1986).
From this point on, the wealth of images revealed by increasing magnification of the set’s boundary (that is, by increasing resolution in the calculations) is almost impossible to describe. Such a journey into the Mandelbrot set, seen best on videotape, is an unforgettable experience. 36 As the camera zooms in and magnifies the boundary, sprouts and tendrils seem to grow out from it that, upon further magnification, dissolve into a multitude of shapes— spirals within spirals, seahorses and whirlpools, repeating the same patterns over and over again (figure 6-20). At each scale of this fantastic journey—in which present-day computer power can produce magnifications up to a hundred million times!—the picture looks like a richly fragmented coast, but featuring forms that look organic in their never-ending complexity. And every now and then we make an eerie discovery—a tiny replica of the whole Mandelbrot set buried deep inside its boundary structure.
Since the Mandelbrot set appeared on the cover of Scientific American in August 1985, hundreds of computer enthusiasts have used the iterative program published in that issue to undertake their own journeys into the set on their home computers. Vivid colors have been added to the patterns discovered on those journeys, and the resulting pictures have been published in numerous books and shown in exhibitions of computer art around the world. 37 Looking at these hauntingly beautiful pictures of swirling spirals, of whirlpools generating seahorses, of organic forms burgeoning and exploding into dust, one cannot help noticing the striking similarity to the psychedelic art of the 1960s. This was an art inspired by similar journeys, facilitated not by computers and the new mathematics, but by LSD and other psychedelic drugs.
The term psychedelic (“mind manifesting”) was invented because detailed research had shown that these drugs act as amplifiers, or catalysts, of inherent mental processes. 38 It would seem therefore that the fractal patterns that are such a striking characteristic of the LSD experience must, somehow, be embedded in the human brain. The fact that fractal geometry and LSD appeared on the scene at roughly the same time is one of those
amazing coincidences—or synchronicities?—that have occurred so often in the history of ideas.
The Mandelbrot set is a storehouse of patterns of infinite detail and variations. Strictly speaking, it is not self-similar because it not only repeats the same patterns over and over again, including small replicas of the entire set, but also contains elements from an infinite number of Julia sets! It is thus a “superfractal” of inconceivable complexity.
Yet this structure whose richness defies the human imagination is generated by a few very simple rules. Thus fractal geometry, like chaos theory, has forced scientists and mathematicians to reexamine the very concept of complexity. In classical mathematics simple formulas correspond to simple shapes, complicated formulas to complicated shapes. In the new mathematics of complexity the situation is dramatically different. Simple equations may generate enormously complex strange attractors, and simple rules of iteration give rise to structures more complicated than we can even imagine. Mandelbrot sees this as a very exciting new development in science:
It’s a very optimistic conclusion because, after all, the initial meaning of the study of chaos was the attempt to find simple rules in the universe around us. . . . The effort was always to seek simple explanations for complicated realities. But the discrepancy between simplicity and complexity was never anywhere comparable to what we find in this context . 39
Mandelbrot also sees the tremendous interest in fractal geometry outside the mathematics community as a healthy development. He hopes that it will end the isolation of mathematics from other human activities and the consequent widespread ignorance of mathematical language even among otherwise highly educated people.
This isolation of mathematics is a striking sign of our intellectual fragmentation and as such is a relatively recent phenomenon. Throughout the centuries many of the great mathematicians made outstanding contributions to other fields as well. In the eleventh
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century the Persian poet Omar Khayyam, who is world renowned as the author of the Rubaiyat, also wrote a pioneering book on algebra and served as the official astronomer at the caliph’s court. Descartes, the founder of modern philosophy, was a brilliant mathematician and also practiced medicine. Both inventors of the differential calculus, Newton and Leibniz, were active in many fields besides mathematics. Newton was a “natural philosopher” who made fundamental contributions to virtually all branches of science that were known at his time, in addition to studying alchemy, theology, and history. Leibniz is known primarily as a philosopher, but he was also the founder of symbolic logic and was active as a diplomat and historian during most of his life. The great mathematician Gauss was also a physicist and astronomer, and he invented several useful instruments, including the electric telegraph.
These examples, to which dozens more could be added, show that throughout our intellectual history mathematics was never separated from other areas of human knowledge and activity. In the twentieth century, however, increasing reductionism, fragmentation, and specialization led to an extreme isolation of mathematics, even within the scientific community. Thus chaos theorist Ralph Abraham remembers:
When I started my professional work in mathematics in 1960, which is not so long ago, modern mathematics in its entirety—in its entirety—was rejected by physicists, including the most avant- garde mathematical physicists. . . . Everything just a year or two beyond what Einstein had used was all rejected. . . . Mathematical physicists refused their graduate students permission to take math courses from mathematicians: “Take mathematics from us. We will teach you what you need to know. . . .” That was in 1960. By 1968 this had completely turned around. 40
The great fascination exerted by chaos theory and fractal geometry on people in all disciplines—from scientists to managers to artists—may indeed be a hopeful sign that the isolation of mathematics is ending. Today the new mathematics of complexity is
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making more and more people realize that mathematics is much more than dry formulas; that the understanding of pattern is crucial to understand the living world around us; and that all questions of pattern, order, and complexity are essentially mathematical.
PART FOUR
The Nature of Life
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