The Fractal Dances of Nature

When a certain operation is applied to the numbers, the ones outside the set flee to infinity. The numbers inside remain to drift or dance about. Close to the boundary minutely choreographed wanderings mark the onset of the instability. . . .

"Things go on this way forever, infinitely various and frighteningly lovely."

Kenton Phillips, a graduate student in astronomy at Penn State, began plotting Mandelbrots on the department computer shortly after those words appeared in the August 1985 Scientific American 'Computer Recreations." Or perhaps it was the pictures that attracted him, pictures more marvelous than slides of our sun's corona, pictures that outdid, for sheer amazement and delight, the finest Op-Art spirals. The pictures were "Mandelbrots," images of the Mandelbrot set, the latest visualization of a ten-year-old math called "fractal" geometry.

Phillips and my friend George Weaver, an astronomy research aide in the office next door to Phillips's, were now setting up the eight megabyte Ridge 32 minicomputer to "walk me through some fractals"—I too had seen the pictures; I too wanted to know how to make them—but Phillips was going to start me out slowly, in black-and white, with a Koch curve, a very simple "scaling" fractal, one that would clearly illustrate the fractal's characteristic irregularity and self-similarity.

"Self-similarity? It's like that old song," explained Weaver, "'There's a log in the bog and a frog on the log.'" Weaver: a reddish beard, twinkling eyes; has a six-year-old son and a baby daughter; is preparing to go to Australia for several weeks to help launch the satellite carrying Penn State's contribution to the supernova watch.

The terminal blinked on: an equilateral triangle, pointing up. Pause. A six-sided star. Pause. With the next image, I ran out of words—names have not been given to these shapes. One was sort of cog with pointed teeth. Maybe. The next was a sand bur. The last three could only be called snowflakes.

Phillips tapped on the keyboard to rerun the program. A slender young man with a great, pale, sloping forehead and a smile that fits his serious features like a party hat. His doctoral project: to calculate what the light from glowing interstellar gases being sucked into black holes looks like—theoretically, the easiest way to "see" a black hole. He draws fractals for recreation. "What I did," he explained, "was write up a program that takes an initial geometric figure—this triangle, that's the 'initiator shape'—and does something to it, what's called a 'scheme of generation.' In this case the scheme is to define each line segment and sick a triangle—a scaled-down version of the generator shape—in the middle of it. The program calculates how to do that for smaller and smaller line segments at all different angles, then connects the dots. There's an arbitrary number of dots I can max out at—for this one, it's 25,000." He handed me prints of the last image in his series (it was a little blurred—the terminal wanted to draw line segments smaller than the width of its pixels) and the third, the pointy cog. "This is what they mean when they say a scaling fractal is self-similar. On the largest scale, these 2 pictures look different, but on smaller scales, they're similar."

Weaver ran his finger around the two images' perimeters. "You see? You have a big bump with two little bumps, a big bump with two little bumps."

Scientific American's fractal illustrations first appeared in an art exhibit called "Frontiers of Chaos" sponsored by the Goethe Institute. The "artists," Heinz-Otto Peitgen and Peter H. Richter, a pair of German scientists, have since published a book, The Beauty of Fractals. Many of the book's images show the edges of the Mandelbrot set. "We discover," write Peitgen and Richter, "a fantastic world there. . . .

"But this is not a familiar experience, that diversity flourishes particularly well at boundaries? Clear contours are the exception in the tension between opposing principles. Every larger conflict reveals a thousand smaller ones."

When Peitgen and Richter's work is criticized as not being "Art" (it lacks "an element of choice," it lacks "an ingredient of human concern"), they reply by quoting Picasso: "Art is a lie that lets us recognize the truth." It may not be a deep truth," they write, "to assert that our world is nonlinear and complex; everyday experience has never taught us otherwise. Yet physics and mathematics, and other sciences following them, have successfully managed to ignore the obvious. . . .

"It came as a surprise even to physicists that there is chaos in their simplest equations."

Phillips smiled, looked at Weaver mischievously. "And now," he said, leading the way down the hall to the Charles River Data Systems 68000, with its color monitor, "for something completely different."

Weaver laughed obligingly. "Saw that movie too, huh?"

Phillips dropped the ham from his voice. "This is the Mandelbrot set." The computer started drawing the familiar pin-headed, arachnoid outline of the Mandelbrot set, with its aureole of colors. The picture, Phillips explains, is produced by putting complex numbers into a simple equation. A complex number can be thought of as representing a point on a graph and so has two parts, one for its north-south distance from the point zero (this is the "real" part), and one for its east-west distance from zero; the east-west distance (the "imaginary" part) carries the notation i, which becomes important when two complex numbers are multiplied, since i squared equals negative one. To draw the Mandelbrot set, then , said Phillips, you simply take a complex number, square it, and add a constant: In complex notation, the equation is z prime equals z squared plus c; Phillips referred to it as "the black box."

"First you put zero plus zero i in there," he said, "square it, and add the constant. You don't plot the result when it comes out, you look at it, you put it back into the black box, you see what happens in the long run, in 500 or 1,000 iterations. For instance, if you took a real number, like one-half , and squared it over and over, you'd get one-fourth, one-sixteenth, and in the long run it would go to zero. A number greater than one will go to infinity. With complex numbers, it's more, er, complex. Some numbers will blow up—"

"That means," interrupted Weaver, "—they tend toward infinity."

This first image, of the complete Mandelbrot set, was now fully drawn and splendidly colored, with swirls and curlicues like colored heat waves peeling off the burning mathematical spider. The constant, c, Phillips was saying, represents the location of each pixel on the computer terminal screen. In order to create this picture 500 pixels high by 500 pixels wide, then, Phillips had to repeat the iterations of z prime equals z squared plus c for 250,000 values of c.

"What you get, finally," continued Phillips, "is a bunch of numbers, c, and what happens to z for that value of c. Now, the exciting part is, What do we do to get the pretty pictures? First we ask, How fast does z blow up? For numbers our here"—he pointed to the white margin of the picture, nearing the pixel called two plus 2 i—"it blows up very fast. For numbers closer to the Mandelbrot set, it takes longer. They diverge more slowly.

"There's a theorem that says once the value of z gets greater than two, it will eventually blow up. But it can take you a long time to see if it goes over two—it can take forever. At some point, you have to give up, you have to decide how many iterations you're going to check. Now, we're not interested in math, we're interested in pretty pictures, so we decided to stop checking at 500 iterations. If it's still under two after 500 iterations, it's in the Mandelbrot set and we call it black. But if it does go over two, we ask, How many iterations did it take to go over two? To that number, the number of iterations, we assign a color. We set up contours. If it takes between one and ten iterations, it's white. Between ten and 20, it's red. And so on.

"You do get very interesting patterns," Phillips continued, "at least what the human eye and brain recognize as patterns." He tapped a key and the Mandelbrot set blinked off; the computer began painting a multitude of multicolored spirals. This was a closeup, he said, a zoom-shot of the edge of the Mandelbrot set right where the head of the spider connected to its body. For the whole picture, he explained, the constant c can have any of 250,000 values with a real part between negative two and positive two and an imaginary part between negative two i and positive two i; to zoom in for this particular shot, he found 250,000 values with real parts between negative 1.254 and negative 1.253 and imaginary parts between 0.22 and 0.33—a dimension not four units high by four wide, but a pinprick in that square, to the northwest of zero and measuring .001 by .001. "This is my favorite of all the regions I found," said Phillips. "There's all these spiral galaxies." Some astronomers have actually begun using fractals in their research, he added, since chaos is related to turbulent flows which are related to the way cosmic gas clouds collapse into intricate stars. He tapped the keys and other closeups flashed past. A knot in a filament turned out, on further magnification, to be a miniature Mandelbrot set, not an exact replica, but closer than my eyes could distinguish: So even the massively complex Mandelbrot set, I remarked, while not as obviously so as a simple "scaling" fractal, was self-similar.

"It's almost like you have this wild piece of modern art," said Weaver, "and you're taking cross sections of it." He elbowed Phillips. "Is that a good analogy?"

Phillips' eyes were glued to the screen. "Sort of," he said, "but you have to go up another dimension. It's got four dimensions—c and z—initial are both complex. You slice through parallel to the z plane and you get the Mandelbrot set. Slice through parallel to the c plane and you get what's called a Julia set. They have pictures of the whole big object, what it must look like—that is, they have three-dimensional cross-sections of the four-dimensional object."

"That's right, I've seen them. It's kind of like a potato with a whole lot of ugly eyes."

"I think it looks more like a rutabaga."

"But it's like opening a geode. You see all the beauty inside."

The poet wrote that Euclid gazed at beauty bare, but the full and continuing appreciation of the beauty of Euclid demands hard and long training, and perhaps also a special gift. To the contrary, it seems that nobody is indifferent to fractals.

In 1975, ten years before he wrote those lines dismissing Euclidean geometry, mathematician Benoit Mandelbrot, then at IBM's Thomas J. Watson Research Center, coined the word "fractal" from the Latin fractus, "broken" or "irregular." The concept had actually been introduced by Felix Hausdorff 60 years earlier, I learned from mathematician Steve Armentrout, who teaches a seminar on fractals for Penn State faculty and students ("One of the real concerns I have," Armentrout told me, "is that there are scads of people trying to use fractals in their work and they don't know enough about it. If you're trying to model a snowflake using fractals, you may not be doing something sensible.). Hausdorff worked in the mathematical field of measure theory. He is known for "Hausdorff's Miracle," an equation which allows you to "measure" a set and assign it a single, characteristic number ("Hausdorff said it was like a dimension," said Armentrout, "so we call it a dimension."). In the 1960s, Mandelbrot was working with equations that earlier mathematicians had considered "exceptional," "pathological," "a gallery of monsters." instead of confirming their monstrosity, however, Mandelbrot, applying what he calls his "intuitive understanding" of Hausdorff's dimension, found the shapes these equations described mimicked such phenomena as the fluctuations in water level of the Nile River. "Shapes which are not fractal," he soon determined, "are the exception." He told one interviewer, "I love Euclidean geometry, but it is quite clear that it does not give a reasonable presentation of the world." And he wrote in Fractal Geometry of Nature, "In brief, I have confirmed Blaise Pascal's observation that imagination tires before Nature: 'L'imagination se lassera plutot de concevoir que la nature de fournir.' "

Armentrout is not so enthralled. "I think Mandelbrot kind of went overboard, so to speak." Armentrout, though relaxed and gracious, has a stentorian voice, and you rather doubt him when he says that anybody, a bright high-school student, could understand the text for his fractals seminar, even when he assures you that you don't need a lot of mathematical background, it's mathematically self-contained. His own specialty is not fractals, but the topology of three-dimensional manifolds.

"I think at this stage," he continued, "it's descriptive, not predictive—and I have my doubts, I wonder how far in that direction you can take it. A lot of us think computer graphics may be one of the major applications of fractals—it's hard to imagine that fractals would be the subject it is today if we didn't have the computer to generate all these pictures.

"Whether fractals are worth a lot of serious attention by anyone other than a mathematician, though, is hard to say."

He leaned back, ran a hand over his head. "We all went through a funny experience a few years ago. There was a mathematical concept called 'catastrophe theory.' Its proponents said it would revolutionize everything, from how you predict the stock market to war. And it collapsed. It didn't live up to its promise. People oversold it. They used catastrophe theory to predict aggression in dogs. It was funny. It was hoopla. Now it's a dead subject. That was the biggest catastrophe about the catastrophe theory."

He leaned forward and rested his arms on the copy of The Beauty of Fractals he was lending me. "A lot of us hope that won't happen to fractals. It has very powerful descriptive capabilities."

Mandelbrot defined a fractal as "a mathematical set or object whose form is extremely irregular and/or fragmented at all scales." When a fractal is magnified (as when the computer lets you zoom in on the Mandelbrot on its screen), those squiggles and spirals and jetting curls refuse to resolve into smooth lines and curves but instead are revealed to be coated with miniature replicas of themselves—the fractal is complexity ad infinitum, or, as Mandelbrot described the set named for him, "unending filigreed entanglement."

As a metaphor for the real world, fractal geometry can describe such complicated, irregular, fragmented, indefinable chaotic shapes as the silhouettes of mountain ranges, clouds, trees, fern leaves, fires; protein surfaces, the effects of acid rain, the distribution of matter in the universe, turbulence: any shape that defies the smooth circles, cylinders, and cones of normal, Euclidean geometry. The coastline, the edge of the ocean and land, is fractal; on the edge of the ocean of life, does death draw a straight line? "The straight line," wrote Friedensreich Hundertwasser (Austrian artist, 20th century), "is something cowardly drawn with a rule, without thought or feeling; it is the line which does not exist in nature." Said Richard Bentley (English scientist, 17th century), "All pulchritude is relative. . . . We ought not. . . . believe that the banks of the ocean are really deformed, because they have not the form or a regular bulwark; nor that the mountains are out of shape, because they are not exact pyramids or cones; nor that the stars are unskillfully placed, because they are not all situated at uniform distance. These are not natural irregularities, but with respect to our fancies only; nor are they incommodious to the true uses of life and the designs of man's being on earth."

Science writer Dietrich Thomsen, in an essay on fractals in Science News, wondered "whether we are not finally taking a very large step beyond Hellenic heritage that has dominated our thought processes for millennia." The ancient Greeks, presumably Euclid among them, Thomsen wrote, "were people who believed very strongly in the power of the human mind to determine what things ought to be. Plato had the nerve to define God; a mind with that kind of chutzpah has no problem telling us what stars and planets ought to be. Some of his inheritance seems to have been more of a hindrance than a help to the development of science." Although science, as a child, needed the straightforward rules of Euclidean geometry so as not to be immediately overwhelmed, Thomsen says, "There have always been large tracts of science where these simple analytic methods were just too complex. Over them people waved their hands in frustration and made qualitative theories, or grossly approximate theories, or no theories at all. It is in these realms that fractals are finding application after application."

"Scientists will (I am sure), wrote Mandelbrot, "be surprised and delighted to find that not a few shapes they had to call grain, hydralike, in-between, pimply, pocky, ramified, seaweedy, strange, tangled, tortuous, wiggly, wispy, wrinkled, and the like, can henceforth be approached in rigorous and vigorous quantitative fashion."

Jack Mecholsky, an associate professor of ceramic science and engineering at Penn State, wants to use fractal geometry to characterize ceramics and other brittle materials.

A tall, engaging man with a polished bald head, bearskin-like beard, and ink-spot eyes, his face washed with disappointment when I directed him to skip the basics of what fractals are and just zero in on how he applies them. He could not concede. It's just too exciting. Fractal geometry, he said, is probably the most useful math ever invented.

"Euclidean geometry, the geometry you learn in high school," he began, "accounts for the very plain shapes, squares, circles, anything that's smooth, and it gives us the relationships between the shapes' geometric parts—for example, the area of a circle is related to its perimeter by a known constant. But what about shapes that aren't so mathematically rigorous? Suppose we give this circle—" he brusquely cleared a patch of blackboard and chalked a circle—a little roughness, a squiggle?" He overlaid a serpentine, crinkled line, a lasagna noodle, on his smooth white circle. "The perimeter has to take the squiggle into account, it was to get longer. The area, on the pother hand, would change very little, in fact, if we're careful with our drawing, it might not change at all. So the perimeter, P, is no longer proportional to the area, A, to the one-half power." He wrote on the board "P ~ A to the ½ and crossed it out. "Euclidean geometry fails.

"But fractal geometry lets us define P." He wrote on the board, reading along, "P is proportional to A to the power of D—the fractal dimension—over 2. You see, fractal geometry allows for dimensions that nonintegers—fractions—dimensions that exist somewhere between a one-dimensional line and a two-dimensional plane.

"Here's a Euclidean line—" he drew a straight line between two points marked A and B. "And here's a fractal line." He scribbled over top of the straight line, drawing something like the squiggle of an EKG. The length of the Euclidean line, he explained, is the same no matter how you measure it. The fractal line, on the other hand, seems to grow longer as you use smaller and smaller measuring sticks: If you use a rod as long as the distance from A to B, the length of the fractal line will be one. But if you measure the line with tiny discs, 20 of which make up one rod, and cover the line completely with a few of these discs as possible, then the length of the fractal line will not be 20 discs, but many more, for the discs can and must follow the serrated contour of the line. The difference, fractal enthusiasts like to say, is the same as the difference between measuring the coastline of Britain as drawn on a map, and measuring it by walking at the waterline: out around a projecting rock, in where a storms's scooped an armload of sand, out again along the mud flats because the tide is low. . . . "The fractal dimension of this line," said Mecholsky, plunking his chalk in to the blackboard tray, "might be 1.2, for example. The fractal dimension is an indication of how much wiggle is there—the more wiggle, the higher the number.

"You can also have a dimension between two and three. Here's an analogy: Take a sheet of rubber, like a broken balloon, and stretch it taut, then poke your fingers up from underneath. It's sort of three dimensional, but most of it's really only two dimensional. The fractal dimension would be a number between two and three, say 2.2."

In 1976, Mecholsky, then at Sandia National Labs, heard a talk on the fractal nature of metal fracture surfaces by Dann Passoja, at the time working at Union Carbide and now an adjunct professor at Penn State. Fractals impressed Mecholsky as "intriguing" and "potentially very useful," and when he moved to Penn State in 1984, he suggested that an undergraduate tackle a fractals project for her senior thesis in ceramic science. He, Passoja, and Karen Feinberg (now Karen Feinberg Ringel and a graduate student at Georgia Tech) took samples of brittle ceramics, broke them, and measured the fractal dimension of the fracture surface. They found that the toughness of the material was proportional to the fractal dimension, that is, K subscript Ic ~ D* ½, where K is toughness; the subscript Ic identifies the type of stress on the sample (Mode I loading) and the value at which the sample breaks (the critical value); and D* is the decimal part of the fractal dimension. In other words, the graph of toughness plotted against roughness is a line with slope ½.

Enthralled, Mecholsky applied for funding to bring a graduate student onto the project. Enter Tom Mackin: He earned a B.S. in engineering science and mechanics (Penn State, 1980); worked at Johns Hopkins briefly; took off for Ocean City, Maryland, where he wrote poetry and science fiction stories and tried to start a health club; returned to Penn State for a master's degree. Mackin has the buoyancy and athletic build of someone you'd expect to run a health club; giving a seminar on fractals as part of his thesis defense, he showed the relaxed, jovial stage presence of a rabbit-in-the-hat magician performing for children. He and Passoja got on famously. Passoja had quit his job at Union Carbide and set himself up as an artist, a painter, in Manhattan. A member of Mackin's thesis committee, Passoja lent him books on Einstein with his own calculations on the fly leaves. "He's so intelligent," said Mackin, describing him to me one hot afternoon, his cubicle filled with dust from a missing wall (renovation was in progress), "that you're not sure he's not crazy. He's probably too bright—he makes connections you can't understand. It took me a year to understand how his brain works, now we have great conversations. He sends me equations and proofs to check. He's a very creative scientific thinker. I'm a little more careful mathematically." Mackin's response to Feinberg's thesis was, he recalled, "That's terrific. Let's get more samples."

Magnified, the broken ends of brittle materials resemble mountain ranges. Mackin's sample mountains were different kinds of alumina, a hard, white substance used for sparkplugs, crucibles, microelectronic substrates, and labware; several glass ceramics, from which rocket nose cones and ceramic stovetops are made; and zinc selenide, laser window material. He took a fingernail-sized fragment of each, sank it in epoxy, polished it until the top of the highest mountain peak appeared, now looking like an island in the epoxy sea, photographed that island, polished some more, photographed again. He was, in effect, taking several cross-sections of the fracture surface and ending up with fractal curves—just as when, on the computer, you take a cross section of the four-dimensional Mandelbrot rutabaga, and end up with a fractal Mandelbrot or, if you slice it in the other direction, a Julia set. Analyzing the perimeters and areas of these islands, using an "entirely new and very powerful" technique Passoja and Mandelbrot had developed, gave Mackin the samples' fractal dimensions; he double-checked the number by calculating the fractal nature of the mountain ranges' silhouettes, using an equation Mandelbrot had recovered from little-known work by the English meteorologist and turbulence expert Lewis Fry Richardson (Mandelbrot calls him "a great scientist whose originality mixed with eccentricity") in 1961.

But when Mackin plotted his fractal dimensions against K subscript Ic, looking for the magical slope ½, he recalled, "I had data points all over the place. So I thought, was it totally fortuitous? Did they luck out with that slope ½? Or was slope ½ correct? Maybe we can order the data. That's when we saw there were families of lines, all with slope ½." To check that result, Mackin took a number of samples of chert, a flint American Indians found was very difficult to shape into arrowheads until after they heated it. Mackin's samples were heat-treated at Sandia National Laboratories, were Mecholsky and Passoja got most of their samples, according to a strict regimen that would emphasize chert's geometric toughening mechanism. When Mackin calculated the cherts' fractal dimensions and plotted those numbers against toughness, the line had a slope ½. The families of lines wasn't a fluke. "It's very hard to say at this point why some materials should lie along the same line. What we have is an empirical relationship—it's all very speculative. But that's the way original work is. You have to start somewhere. You just try.

"What we need now is a theoretical argument that leads to this relationship.

And that might not be possible. You see, now we have the fractal dimension of a contour line of the fracture surface—everything that's at a certain elevation—and it's 1.12 or 1.31 or something. But that number doesn't tell you anything about the two-dimensional plane. The real fractal dimension of the surface should be between two and three." in Mecholsky's analogy, then, a fracture surface isn't a wiggly line, it's a rubber sheet with your fingers poking up through. "Now, Passoja and Mandelbrot wrote a paper saying that the real fractal dimension of a surface like this is simply one plus the one-dimensional fractal dimension." If the rim of the island is 1.12, then the mountain range is 2.12. But I don't know. I can't see it. I've discussed it with Steve Armentrout, and he can't see it either. These contour lines are very interesting, strange little beasts."

Part of the problem is that fractal geometry, like any mathematical theory, can only describe nature through analogy. Passoja, at Mackin's fractal seminar, pointed out that the fractal dimension is statistical: "We have these very rigid models in mind, and they don't apply." In a 1984 paper in Nature, he, Mandelbrot, and Alvin Paullay of Columbia University explain, "The term 'fractal' was chosen in explicit cognizance of the fact that the irregularities found in fractal sets are often strikingly reminiscent of fracture surfaces in metals (though not, for example, in glass). However, metal fractures are only extremely crinkly (down to the limits of their microstructural size range), while fractals are infinitely crinkly. Hence, it is not possible to say that metal fracture is strictly a fractal.

"Nevertheless, metal fractures resemble a fractal so closely that it makes good sense to use a fractal for modeling a metal."

Or, as Mecholsky put it, "There's nature, and then there's math."

If the theory does prove true, if toughness is related to roughness through the material's fractal dimension, if the fracture surface is fractal, it ought to show the characteristics of a fractal: irregularity (and obviously it does) and self-similarity.

"That's why we're so excited about this," said Mecholsky. He had returned to his office chair, behind a desk cluttered with samples of brittle ceramics: a crucible, an alumina chess set, dime-sized epoxy plugs with ceramic centers looking something like Goetz's caramel creams. It's like we've been looking at it for a long time without seeing it."

He handed me an article from the Proceedings of the Materials Research Society's 1986 meeting: "Crack Propogation in Brittle Materials as a Fractal Process," by Mecholsky, Mackin, and Passoja. If indeed a fracture is a fractal process, I read, then several conclusions can be made. There should be an initiator of fracture and a scheme for generation. If these exist, as they should for a fractal process, do they exist down to the atomic scale? Further, we should be able to relate atomic processes to macroscopic processes.

"Normally on the atomic scale," Mecholsky was saying, "we talk about energy. On the macroscopic scale, we talk about geometry—things on the surface that we can see to measure. So we have to make a switch, on some scale, from energy to geometry."

The atoms in a hard material, he explained, are thought of as being arranged in