Why Study Fractals? by Joe
Pagano
Mathematics has a way of taking us by the hand and
not just leading us down the path of reason, as Pythagoras once
said, but sometimes down the path of insanity. With all the beautiful
truths that math can show us, there are also inherent contradictions
of nature that this field forces upon our senses. Such seeming impossibilities
are found within the world of fractals, those weird yet curious
geometric objects that have caused us to look at nature in a whole
new way. From the surface of a mountain to the head of a broccoli,
fractals are being used to explain things that we normally take
for granted.
Fractal comes from the Latin word for broken and was
coined by the mathematician Benoit Mandelbrot in 1975. The reason
for the name is that fractals when viewed closely are self-similar;
that is if we break off a part of the fractal, we essentially have
the same shape, albeit a smaller piece. Fractals are generated by
a procedure called recursion in mathematics. In lay terms, this means
that we start a process according to a specific rule and then let
this process continue forever. To understand what this means, let's
take a specific example which will also generate a very famous fractal
called the Koch Snowflake, so named after a Swedish mathematician.
This fractal demonstrates the insane and curious world of fractal
geometry.
To generate the Koch Snowflake, we start with an equilateral
triangle. Now along each side, we construct another equilateral triangle
starting a third of the way from each vertex. To make this easier
to visualize (see here Koch Snowflake) take an equilateral triangle
of side length 1. Then on each side length, we omit the middle 1/3
of the segment and construct another equilateral triangle, the sides
of which are all length 1/3, starting from the ends of the deleted
portion. This completed, we have the second step or iteration of the
Koch Snowflake. We then do this again, only now we have more sides
to work with. Proceeding this way, we end up with successive iterations
of the Koch Snowflake. Notice that if we were to circle any region
of this curious shape, we would have a self-replicating pattern, and
the seed from which the curve could continue to grow.
Now where the insanity comes in is that this particular
fractal illustrates the bizarre reality of a geometric shape which
has an infinite perimeter, yet finite area! How strange indeed this
world of fractals is. Moreover, fractals illustrate the concept of
non-integral dimension. That is, once we enter this phantasmagorical
world, ordinary dimensions like 1, 2, and 3 (our world is three-dimensional--4
if we think like Einstein did in terms of space-time) are no longer
appropriate as we can find fractals with dimensions like 1.3. In fact,
the Koch Snowflake has dimension 1.26!
Tres outre, this world of fractals. So why bother with
them? Well, according to Benoit Mandelbrot, the mathematician who
coined the term fractal, most of the shapes in nature have bizarre
non-integral dimensions like those typified by these weird fractals.
And since nature is all around us, it might behoove us to consider
the ramifications of these insane dimensions. Think about this next
time you munch that head of broccoli.
See more at About Joe Pagano and Poems on Fractals
About the Author
Joe propagates his teaching philosophy through his articles and books
and is dedicated to helping educate children living in impoverished
countries.
