by Scott Camazine
New ways of looking at the world help explain the development of complex and beautiful patterns in nature
A hike in the woods or a walk along the beach reveal an endless variety of forms. Nature abounds in spectral colors and intricate shapes — the rainbow mosaic of a butterfly's wing, the delicate curlicue of a grape tendril, the undulating ripples of a desert dune. But these miraculous creations not only delight the imagination, they also challenge our understanding. How do these patterns develop? What sorts of rules and guidelines shape the patterns in the world around us?
Some patterns are molded with a strict regularity. At least superficially, the origin of regular patterns often seems easy to explain. Thousands of times over, the cells of a honeycomb repeat their hexagonal symmetry. The honeybee is a skilled and tireless artisan with an innate ability to measure the width and to gauge the thickness of the honeycomb it builds. Although the workings of an insect's mind may baffle biologists, the regularity of the honeycomb attests to the honey bee's remarkable architectural abilities.
Although some of nature's artistry is no longer a mystery, other patterns are more subtle and perplexing. They may possess a mathematical regularity, but that does not help explain how they form. Consider the Fibonacci sequence, named after the medieval Italian mathematician Leonardo Fibonacci. Begin with 0 and 1. To obtain each succeeding number in the series, simply take the sum of the previous two numbers. The result is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,....and so on. The sequence seems to be nothing more than the idle doodling of a mathematician in a daydream. Yet for reasons not fully known, nature has incorporated Fibonacci's sequence into many of her botanical blueprints. Look at a daisy and notice the pattern of the florets. They are arranged as two sets of logarithmic spirals, intertwined. Each floret belongs to both a right-handed and a left-handed spiral. Now, carefully count the number of clockwise and counter-clockwise spirals and you will find that they are consecutive Fibonacci numbers. Though scientific controversies still exist over why such a scheme is found in daisies, pine cones, pineapples and other whorling botanical structures, their artistic beauty remains unquestioned.
Simple computer models yield elegant patterns
Next consider seashells, so often decorated with bold patterns of stripes and dots. Biologists seldom gave much thought to how these mollusks create the beautiful designs that decorate their calcified homes. Perhaps they simply assumed that the patterns were precisely specified in the genetic blueprint contained in the mollusk's DNA. But some years ago, scientists, skilled in both biology and computer science, began to look at pattern formation in an exciting new way. One of the first things they realized was that two individuals of the same species were similar, but not identical. Like the fingerprints on one's hand, they are alike yet not alike. This simple observation led them to hypothesize that the patterns on shells, the stripes on a zebra, and the ridges on our fingertips are not rigidly predetermined by the genetic information inside the cell's nucleus. Organisms are not built as a house is built, by meticulously following an architect's plans.
Instead, genes appear to take a more generalized approach, specifying sets of basic rules whose implementation results in organized form and pattern. Tackling the problem of how markings develop on shells, these scientists proposed a few simple rules for how pigment precursors in cells might diffuse along the snail's mantle at the growing edge of the shell. Then, by repetitively implementing these simple rules in a series of computer simulations, they "created" shell patterns with a startling similarity to real shells. These scientists readily admit that this similarity does not prove that shell patterns develop in the manner they hypothesize, but it does suggest that simple mechanisms could account for some of the complex and varied patterns observed in nature.
Over the years, these same ideas have been applied to many questions in developmental biology concerning how structures become organized. One of the greatest biological mysteries yet to be solved is how a single egg --— apparently devoid of structure — becomes a child. The human cell does not contain enough information to specify the location and connections of every neuron in the brain. Therefore, much of the body's organization must arise by means of more simple developmental rules. In nature many systems display extreme complexity, yet their fundamental components may be rather simple. The brain is an organ of unfathomable complexity, but an isolated neuron cannot think. Complexity results from interactions between large numbers of simpler components. With the advent of powerful computers, mathematicians, chemists, physicists, biologists and even high school computer hackers began to discover how simple interactions between large numbers of subunits could yield intricate and beautiful patterns. Suddenly people were studying all sorts of phenomena both mundane and bizarre — piles of sand, dripping water faucets, slime molds, leopard's spots, forest fires, flocking birds and visual hallucinations. Though these various phenomena have little in common, they are all fertile subject matter for those who study nature's complexity. And this emerging field has given us a new vocabulary including such terms as chaos, fractals and strange attractors.
Some patterns self-organize
The study of complexity provides new insights into how patterns develop in nature. One exciting finding is that order often arises spontaneously from disorder; patterns can emerge through a process of self-organization. One of the best ways to visualize how patterns self-organize is to employ simple computer programs that simulate a natural process. One such category of computer programs are called cellular automata. They are simulations played on the equivalent of a computer checkerboard. In the simplest version, one starts with a single row of cells. In the example above, concerning shells, each square of the checkerboard represents a hypothetical cell along the edge of the snail's mantle. The cell could either produce a color pigment, or none at all. The future state of the cell (whether it produces pigment or not) is determined by the cell's present state and the state of its nearest neighbor cells on either side. One rule for pigment production might be as simple as this: if the cell currently produces pigment and at least one of its adjoining neighbor cells produces pigment, then the cell will continue to produce pigment in the future. The state of the cells changes over time and each row of the checkerboard displays the next step in the process, just as the growing shell displays its developmental history. What is remarkable is that even if one starts out with a completely random array of cells at the beginning, a remarkably organized pattern emerges — order arises from disorder. More complicated cellular automata models have been developed to explain the stripes on zebras, the mottled patterns on fish, the growth of snowflakes, and even clustering of neurons in the brain.
Fractal patterns in nature
Of course, not all patterns in nature are regular. Billowy clouds, flickering flames, lightning bolts, the pattern of veins on a leaf, the architecture of the lung's passageways — these are examples of patterns without obvious regularity. But looks can be deceiving. Many irregular patterns are not simply random. They often display an underlying structure, a kind of regular irregularity that can be mathematically described. Such objects have been called fractals, a term coined by Benoit B. Mandelbrot of IBM's Watson research center meaning broken or fragmented. Fractals are intricate structures that continue to show rich detail no matter how closely one zooms in for a look. England's meandering coastline looks wiggly whether viewed from a plane, while walking along the coast, or up close with a magnifying glass. In contrast, think of a circle. When smaller and smaller portions of a circle are magnified, the segments become straighter and straighter. At higher magnifications, the circle loses detail. But fractals keep on going, repeating similar intricate patterns at many different scales of magnification.
Investigators began to wonder how these fractals form. Two scientists, Thomas A. Witten III and Leonard M. Sander have proposed a very simple mechanism for certain fractal forms. They call the process diffusion-limited aggregation. Imagine sticky particles coming into contact with each other and aggregating to form a cluster. Start with one particle in the center and release another sticky particle which randomly diffuses inward. When the particle finds the one in the center it sticks and stays put. Now repeat the process over and over, thousands of times. A meandering, tenuous cluster will grow. It will be a fractal. With such simple growth rules, these fractals are easy to create on a personal computer, and they resemble examples in nature such as the buildup of soot in a chimney, the path of a lightning bolt as it tears through the sky, or the sprawling radial growth of lichen on the surface of a stone.
Whether regular or irregular, patterns in nature have always delighted naturalists, photographers, and artists. And for those with an inquisitive mind — not content merely to gaze in wonder — nature's complex patterns provide the added attraction of mystery surrounding artistry.