On a recent flight from Oakland to Los Angeles I watched the waves of the Pacific crashing against the shore as the plane gained elevation. After a few minutes of distraction, I again gazed down at the interface of land and water. We were much higher now, well on our way to cruising altitude. I was puzzled when I noticed that although I could still see the long lines of the waves, they seemed motionless, frozen in space. It took a while for me to realize that I was now watching waves on an entirely different scale—these were many miles long: the grandparents of the breakers we’d see if standing on the beach. Those waves now appeared as faint, miniscule ripples between the vast swells that arced and crossed each other, moving too slowly for perception from such a distance.

I understood that this was a dramatic illustration of “fractality”—self-similarity across scales. This principle, first articulated and explored by mathematician Benoit Mandelbrot in the 1950s, is the basis of fractal geometry, a new approach to mathematics that has profound implications and widely diverse applications. Fractal geometry grows out of iteration: the solution of an equation is fed back into the equation as one of its terms, yielding a second solution, which again is fed back, replacing the previous solution, and so on. The development of fast computers and high-resolution graphics has made it possible to calculate large numbers of iterations rapidly and to map them as patterns on a video screen. One highly significant attribute of fractal geometry is that it can come very close to describing the complex or chaotic forms found in nature—far closer than Euclidean geometry, which has trouble with forms that require more than straight lines and simple curves.

One of the standard models of fractal geometry in nature is a coastline. Whether viewed from a satellite orbiting Earth, a jetliner cruising several miles high, the observation deck of a skyscraper, a blufftop, the rocky edge of a tide pool, or through a magnifier, the place where sea meets land has a similar shape—though generally the closer we look, the more detail we find. As I’d observed, the same is true of the waves of the sea across a wide range of scales.

Once we become aware of this fractal dimension, we can recognize it in many places, and begin to see how it reveals many of the organizing principles of growth and of natural forms. It is easy to note the similarity between a head of broccoli and a leafy tree, a sponge and a cauliflower, or the repeated patterns of fern fronds. Mycologist Paul Stamets recently showed a comparison of a scanning electron microphotograph of a fungal mycelial network to one of an animal’s neural network, and to a projected mapping of intergalactic “dark matter.” Not only did they appear remarkably similar, but within a day I’d come across a photograph of lightning activity that fit the same pattern.

The fact that immensely complex patterns can be generated by iteration of very simple equations has led to many new insights into and understandings of natural processes. The principles of fractal geometry even help us begin to understand such chaotic and extremely variable systems as turbulence, and has applications in economics and other fields. Fractal geometry is at the heart of much computer-generated “natural” imagery, and at the core of some computer image-compression systems.

The beauty of fractally generated images interfaces closely with the world of art. And through its clear connection to the ancient Hermetic axiom “As above, so below,” it can lead us to the realm of spirit, where the wonders of Nature meet the beauty of Science—a most marvelous coastline.

On my way back to Oakland, I intended to watch the waves more closely, to see if I could perceive the shift from one scale to the next. At first I was disappointed—the plane took an inland route! Before long, however, I noticed that the same phenomenon occurred with the rounded, golden California hills. At cruising altitude, the familiar contours of valleys dark with oaks that we see from the ground were minute details that punctuated the larger forms of the ranges, which still appeared smooth, rounded, and golden.

Fractal images generated by Hal Hughes, using Fractint. For more information on fractals and related subjects, and to download Fractint and other free fractal programs, see http://spanky.triumf.ca.