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.
