Fracticals and Human Life

Where in nature do we see straight lines, exact triangles, perfect circles and other standardized shapes? Nowhere. As mathematician Benoit Mandelbrot, a man I will tell you more about in a moment, put it, "Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line." Nature does not follow traditional Euclidian geometry.

Let us think for a moment about the lowly cauliflower, for example. If we want to get some sense of measurement of a cauliflower head, we can easily weigh it and come up with a number. However, if we want to measure its surface, we have great difficulty arriving at a number, for the surface is neither flat nor smooth. To measure it, we would need some way to measure irregularity or roughness. What is more, if we cut off one of the florets and study it, we see that it, like the whole cauliflower head, is also rough. The same is true if we break a sprig off the floret - and is even true of smaller pieces of the sprig. In fact, each smaller part is like the whole cauliflower, only more diminutive.

It turns out that this principle applies to many things in nature, such as trees. If we look at a tree closely, we see that the individual branches look like small trees, and the same is true of the smaller branches off the larger ones.

Now let us return to the mathematician. Mandelbrot was not the first to notice that in natural formations, small parts often resemble the whole. However, before him, people regarded this feature as an isolated, nonintuitive curiosity. In the 1970s, however, Mandelbrot took this phenomenon - which he calls "self-similarity" - and used it as a basis on which to build a new kind of geometry, a non-Euclidian geometry for applying science and measurement to non-smooth objects in the real world. Self-similarity is the property of having a substructure analagous or identical to an overall structure. For example, a part of a line segment is itself a line segment, and thus a line segment exhibits self-similarity. By contrast, no part of a circle is a circle, and thus a circle does not exhibit self-similarity. Many natural phenomena, such as clouds and plants, are self-similar to some degree. In the process, he coined the word fractals to refer to these irregular shapes. However, more importantly, he demonstrated that the irregularly shaped objects in nature do not have a random shape. Such shapes actually follow simple rules to generate seemingly complex and chaotic patterns. Mandelbrot said the roughness of shapes in nature is not a mess but something in which he found "very strong traces of order." (He developed the word fractals from the Latin fractus, which means broken.)  Mandelbrot went on to write a book about his new geometry based on fractals, which he also described as the "science of roughness." He said he preferred the word roughness to irregularity because grammatically, "irregularity is the contrary of regularity," whereas in nature, the contrary of regularity is rough.

One more example sometimes used to explain fractals is a coastline, which, of course, is irregular or "rough." On a map, we might represent a small section of coastline as a straight line, but, in reality, even small sections do not form straight lines. If we look at that section closely, we see that it is made up of several small peninsulas and inlets. If we look even more closely, we see that each peninsula and inlet has its own bays and headlands. If we continue to look at even smaller sections, we will discover that the pattern is always present. Moreover, the recurring pattern of roughness is more or less the same, no matter how closely we look at the object in question.

Understanding fractals has made possible significant advances in fields as varied as physics, music, linguistics, weather forecasting, medicine, economics and even movie-making. In the case of the latter, for example, a film director needing a shot with a mountain in the background can put into a computer a fractal algorithm of a pattern of peaks and crags, and the computer can generate the whole mountain, reproducing those basic shapes on varying scales. Granted, the result is not a real mountain, but it looks like a real one. In addition, if the director decides the mountain is not rugged enough for the scene, the special-effects people can simply bump up the roughness number and regenerate the mountain.

All of this suggests that nature does have an order, even if the order is not the smooth surface on which Euclidean geometry paints. Rather, the order of nature is more like the rough surface of fractal geometry. If you have followed me so far, maybe we can think of this type of order in nature as a metaphor of the type of order we find in human life. After all, human beings are part of nature. What if we viewed what occurs in the human life as having an order something like that of fractal geometry? Human life is hardly smooth. It has all the twists and turns that we find in nature. In our limited experience of our personal lives or of human life on this planet, it may appear irregular. In reality, it may simply be rough, having an order that one might perceive if one could gain the proper perspective. It has a design, if you please, even if our limited perspective makes it look random.