Why study Fractals?

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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.


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