Welcome to a mathematical language of the
clouds, mountains and rivers
Have you ever looked at all the
perfect geometric shapes in your textbook and wondered to your self, "Hey, I’ve
hardly seen such perfect shapes in nature. Isn’t there some geometry that describes
them?"
Someone did. Benoit Mandelbrot, a French mathematician, made the famous observation,
"Most of nature is very, very complicated. How could one describe a cloud? A cloud is
not a sphere… It is like a ball but very irregular. A mountain? A mountain is not a
cone… If you want to speak of clouds, of mountains, or rivers, of lightning, the
geometric language of school is inadequate."
So what did he do? He went ahead and created a brand new science called fractal
geometry. Trees, clouds, river deltas and the coastlines — they all can be explained
by it. A fractal is a complex geometric object whose component parts seen from up close
resemble the entire structure from far away, meaning it is "self-similar". (see box ).
Today, fractals help classify and analyse order in natural phenomena, such as the
branching of blood vessels, the turbulence in fluids, and the distribution of galaxies in
space. Mandelbrot himself says that fractal geometry deals with an aspect of nature that
almost everybody had been aware but nobody was able to describe in formal mathematical
terms.
Fractal comes from the Latin fractus, meaning broken apart and it was explained in
detail by Mandelbrot’s brilliant book The Fractal Geometry of Nature in 1975.
Here’s a fractal teaser. How long is a coast? There is no clear answer if you use
this new science. Mandelbrot showed that, since the measured length of a coast can be
extended indefinitely by going into smaller and smaller scales, there is no clear-cut
answer to the question. But he defined a number between 1 and 2 that characterized the
jaggedness of the coast, where 1 is the space taken up by a straight line and 2 is the
space taken up by a plane. A jagged line will take up a number in between. This means
smoother the line, the more close to 1 it’ll be. More jagged the line, the more close
to 2 it’ll be. He found this figure to be 1.58 for the British coastline and 1.7 for
the much rougher Norwegian coast (See box ).
The concept of a fractal dimension, which was at first a purely mathematical idea, has
become a very powerful tool for analyzing the complexity of fractal shapes, because it
corresponds very well with our experience of nature.
In so doing Mandelbrot has in a way gone beyond Albert Einstein (who said that time was
the fourth dimension) to discover that his fourth dimension includes not only the first
three dimensions, but also the gaps or intervals between them, the fractal dimensions. The
geometry of the fourth dimension — fractal geometry — is now recognized as the
true Geometry of Nature.
A fractal reveals greater
complexity as it is enlarged, showing existence of
"worlds within worlds"
Self-Similarity It means that on
analysis of a certain structure will bring up the same basic elements on different scales.
Mandelbrot described self-similarity by breaking a piece out of a cauliflower and pointing
out that, by itself, the piece looks like a small cauliflower. He takes a small piece out
of that and so on. Every part looks like a full vegetable. This property is illustrated by
some shapes below, in which every part takes the same shape as the complete shape
 A tree from a box!
Take a box and place two smaller boxes on top of it as shown in the figure. Now take two
even smaller boxes and put them on the two small boxes and so on. After you’ve
repeated this a few dozen times, you get a figure very suspiciously like a tree. You can
change the shape of the tree using different sized boxes.
 Coast in a coast in a coast in a coast in...
A characteristic of objects created from fractals is that they exhibit scale invariant
behaviour. A commonly given example of this is the coast of Norway. If one examines the
coast of Norway, one sees a series of long, twisting inlets called fjords. Closer
examination of these inlets reveals that these have long, twisting inlets of their own. In
fact, the more closely one examines the coastline, the more fjords one sees, each smaller
than the next.
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Former American Vice President Al Gore has found fractals
useful in the way he views the world. He once said, "The ideas in the fractals, both
as a body of knowledge and as a metaphor are an incredibly important way of looking at the
world… it often allows us to look at social and political matters and find ways to
connect the dots that haven't made sense before."
