If you have ever lain on your back to watch big puffy cumulus clouds boiling across the sky, if you have stood on a coastline enveloped in the sight of the ocean swelling and uncoiling breakers against the land, if you have ever contemplated the mountains, then you know.
There’s something revitalizing and deeply fascinating about the recurring and ceaselessly variable patterns of nature. Perhaps we stop to marvel at the way a network of erosion has etched itself into a hillside or, on a vaster scale, sculpted the intricate chasm of the Grand Canyon. Or we pause to appreciate the sensuous angles of tree branches; the exhilarating puffs and bursts of wind on a blustery day; the wild, shifting shapes of fire; the spatters of mold and lichens on the face of a cliff; or a dark night’s crystal scatter of stars.
Nature’s patterns, at once familiar and unexpected, inspire us, satisfy us, sometimes terrify us. Poets, mystics, and everyday
travelers on Earth turn to these patterns for solace, for a sense of continuity, for a glimpse of the divine mystery.
Nature’s patterns are the patterns of chaos. “Fractal” is the name given by scientists to the patterns of chaos that we see in the heavens, feel on earth, and find in the very veins and nerves of our bodies. The word was coined by the mathematician Benoit Mandelbrot and now has wide use in chaos theory, where fractals refer to the traces, tracks, marks, and forms made by the action of chaotic dynamical systems.
Natural fractal forms include the crack in a rock ledge left by an earthquake or frost heave, the dendritic web of a river system, the once-only shape of a single snowflake. Mathematicians have imitated these natural fractals using various kinds of nonlinear (feedback) formulas. Although infinite in their detail, mathematical fractals lack the subtlety of their natural counterparts. Nevertheless, they have brought scientists closer to visualizing the real movement of chaos that makes natural fractals possible.
Natural Fractals and Ourselves on the Coast
The classic illustration of a natural fractal is a coastline. Mandelbrot introduced the idea of fractals in a paper that asked an elegantly simple but fiendishly complex question: “How long is the coastline of Great Britain?” His answer provided some wondrously curious glimpses into the landscape of chaos.
Imagine Britain from a satellite distance—Britain on a world map. Bend a thread around the craggy line of the coast and then hold it against the scale of miles on the map. How long is the coast? The answer seems simple. Now repeat this procedure on a national map that has more detail. On this new map, we see more of the bays and indentations of the actual coast. Measuring our thread against the scale on this map, we find the coastline measures longer. With a highly detailed maritime chart, it measures longer still. Now try it on foot with a piece of rope and a tape measure, making an effort to encompass every twist and turn. What about going down to the twists and turns on the molecular or atomic level?
By this logic, Mandelbrot arrived at the surprising conclusion that the coastline of Britain must be infinite! We might add that not only is the coastline infinite, but because it is continuously being eroded, it is an infinity that is constantly changing! Mandelbrot also discovered that every coastline, from the smallest desert island to the Americas themselves, has the length of infinity.
A coastline is produced by the chaotic action of waves and other geological forces. These act at every scale to generate shapes that repeat, on smaller scales, a pattern roughly similar to the one visible at the large scale. In other words, chaos generates forms and leaves behind tracks that possess what chaos scientists refer to as “self-similarity at many different scales.”
The shape of a particular tree—which is produced by all the interlocked chaotic dynamics of the genetic program in the seed and the flux in the environment, including available sunlight, weather, disease, soil conditions, the position of other trees, and so on—is mirrored at several scales. The trunk forks into branches, the branches fork into smaller twigs. Twigs contain leaves, which themselves repeat the dendritic pattern in their veins. In its largescale shape and in its small details, the tree is a self-similar record of the moment-by-moment, unpredictable flow-through of the chaotic activity that created it and sustains it.
That record contains not only what is similar about the different elements of the tree, it also contains what is absolutely unique about each element and combination of elements. Trees of the same species standing together in a grove each have a uniqueness that makes us stop and say, “Look at that tree over there. Isn’t it beautiful?” In the angles, turns, and rhythms of its trunk and branches, in its patterns of lichens, moss, and disease, in countless other details, we are glimpsing a dynamic picture of the individual tree—and its life in the flux.
For clarity’s sake, let’s stipulate that the term “self-similar” includes this idea of individual differences and uniqueness as well as similarities. As we’ll see, there’s a vast range of fractal self-similarity that is possible both in the forms of nature and in human consciousness. In some fractal forms—particularly the ones generated on computer screens by mathematical formulas—the self-similarity is somewhat mechanical. In other fractals—fractals in nature and art—what is self-similar is infused with what is different in ways that defy description.
A Chan Buddhist text says, “One particle of dust is raised and the great earth lies therein; one flower blooms and a universe rises with it.” The poet William Blake echoes the Zen text with his instruction in “Auguries of Innocence”: “to see the world in a grain of sand, and eternity in an hour.” Fractal self-similarity is the chaos version of this ancient and poetic truth.
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The Aesthetics of Fractals
A fractal aesthetic encourages us to explore the rich ambiguities of metaphorical connections between ourselves and the world rather than remaining only within the categorical abstractions that separate us from that world.
Our primal sympathy and appreciation of fractal forms contains our appreciation of the openness of forms fluctuating on the edge of life and death, living in the flow between structure and dissolution.
The unities we glimpse in fractal patterns aren’t sentimental unities. They aren’t unities that depend on a theory or even religious idea. They may even be unities that unsettle our theories and ideas. We may appreciate the fractal beauty of a war shredded landscape or peer into the mirror of truth while reading a story about the grotesque conflicts of human nature.
Math Fractals
Fractals came to the public’s attention through the stunning abstractions generated on the computer screens by the famous Mandelbrot set. These images are plots of mathematical formulas. Mathematical formulas are, in turn, formalizations of the rules of logic. Applying a simple nonlinear (feedback) formula or algorithm to the numbers in this region and then plotting their behavior as the formula iterates, mathematicians and computer “fractalnauts” can obtain stunning images that have a certain organic quality and a certain quality that resembles art.
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Mathematical fractals are impressive, but after repeated viewing, the freshness of one of these objects fades. This doesn’t happen with the creations of nature, which emerge out of a holistic chaotic process whereby countless “parts” are subtly interconnected—true chaos as opposed to a mathematical simulation produced by repeating an algorithm. Consequently, natural fractals have an individuality, spontaneity, depth, and quality of mystery that no algorithm—even a nonlinear one—can reproduce.
The Art Beyond Fractals: Joining Reason and Spirit
Throughout our history, art has been integral to the human experience of the world. From the time of Ice Age cave paintings through the Middle Ages, art was an expression of our faith that the Universe is spiritually coherent. Indigenous and peasant cultures lived, and many still live, surrounded by everyday objects—pots, knifes, animal skins—adorned with metaphoric selfsimilarities. They lived closer to the chaotic resonances of nature in which the spirit of life was revealed.
At its root, art contains nests of self-similarity. But the selfsimilarities of art, like those of nature, are deeper and far richer than those in the Mandelbrot set. The kind of “fractal” order in art goes far beyond anything mechanical—anything that can be reduced to didactic description. In fact, it’s what defies our description that defines an artwork’s greatness.
Becoming sensitive to an artwork’s self-similarity is a little like becoming attentive to the way birds, squirrels, and chipmunks interact at your backyard feeder. After watching them for awhile you begin to sense that although there are repeating patterns, within these patterns something unexpected and profound is going on that keeps you absorbed.
Nature makes its fractal forms out of matter and energy. The material of art includes human consciousness, as well. Poems, paintings, and concertos are fashioned out of our categories of perception and language. Artists create discord within these categories by using irony, poetic metaphors (where unlike things are said to be the same), simultaneous harmony, and dissonance among musical notes and other analogous techniques. The concords and discords form patterns that are always surprisingly and significantly self-similar and self-different from each other, reflecting the curious mystery of our being in the world.
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Chaos theory isn’t art, but it points us in a similar direction: the direction we find in the healing images of nature, the direction in which lies our effort to contact the secret ingredient of the Universe we call ‘spirit‘.
