Fractal Geometry


Five years ago, a friend of mine generated a drawing on his computer and gave it to another friend as a birthday present. This was my first experience with fractal geometry. Unlike the usual drawings, it was a kind of image where a colorful geometric (or is it non-geometrical, I wasn’t sure) pattern was repeated without any specific beginning or end. Later on, I saw some of my friend’s similar work. He tried to fathom my astonishment while explaining, “It’s a computer program. That’s how I do it.” He gave me this simple explanation without even using the words fractal or Mandelbrot, which I would learn a few years later.

As I said, I would be reintroduced to fractal geometry a few years later. As seen in the picture below, Benoit Mandelbrot couldn’t gain favor with either mathematicians or physicists for some time, even though he succeeded in upsetting the 2,000-year-old Euclidian geometry when he invented fractal geometry. Looking for a definition for his invention, he browsed the pages of his son’s Latin dictionary. He came across the adjective fractus, which means broken or uneven, and this inspired him to name his creations fractals. This was in 1975, eight years after his well-known article on fractal geometry “How long is the coast of Britain?” was published.

With an answer as interesting as its title, this article blazed a trail for mathematicians who later focused on the other epiphanies of fractal geometry. To sum it up, Mandelbrot expressed it like this:

The length you come up with varies depending upon the length of your dividers. If you are measuring with a 100cm divider, then you won’t be able to measure the curves under 100cm. If you are measuring with a much smaller divider, curves under that size will be rounded. The more sensitive your measurements are, the greater your result will be. Ultimately, you will find yourself trying to measure every single grain of sand one after another.

I never wondered how long the coast of Britain (or any other country) was. Even during my most curious period, I knew there were no real men inside the TV, but I was very surprised when I learned how the famous Mandelbrot set, a set that hides more tiny patterns inside each convolution, consisted of a simple equation: Zn+1 = Zn2 + c

As far as I understand, the only trick is that Z0 should be equal to zero and c should be a complex number corresponding to the point of testing. To be more precise, let me explain it this way. We choose a number c and begin the sequence with Z0=0. We multiply Z0 by itself and then add our chosen number c to get the next value Z1. This can be repeated as many times as we want, and the dynamic equivalent of this function becomes the first of the drawings above.

When I did some detailed research, I discovered how mathematics had waited for nearly a century for computers to develop and reach today’s level. This explains why fractal geometry wasn’t accepted at first and why it needed to wait for a clear definition following its emergence. For example, Gaston Julia and Pierre Fatou studied fractals during World War I. When Mandelbrot came across their equations, which were called the Julia set, he understood with the help of his intuition that they could not be explained by Euclidean geometry. He was actually lucky, because he was born in a suitable era for his studies. He also had the opportunity to work with state-of-the-art computers, at least for those days, at IBM. He soon became well known as an eccentric and nutty scientist.

In the light of the knowledge above, even though I find it quite difficult to explain, I can easily conclude that the reason for the success of fractal geometry, which nobody had been able to formulate throughout history, lies in Mandelbrot’s point of view. He also used his point of view in his questions and answers, namely using the scale factor as a main parameter. One other well-known question was “What is the size of a wool-skein?” If we look at it from a distance, it appears to be the size of a dot. When we get closer, we can see strings wrapped over each other. When we look closer still, we see the thin fibers within the string. With a much closer look, the fibers are made of zero-dimensional dots. It is hard to believe that a wool-skein, which you can actually hold in your hand, could be a total size of zero dimensional dots, yet this is the mathematical explanation.

At first glance, it’s also hard to believe that the Koch snowflake consists of triangles repeating themselves on smaller scales as a continuous curve without tangents. However, mathematics has such an unbelievable construction. We start with an equilateral triangle where all three sides are 30cm. On each side, we add new triangles that are one-third the size of the original triangle. If we continue to do this to infinity, we have a snowflake with an infinite side length that covers the area of a maximum concentric circle, just as Helge Von Koch defined in 1904. It all depends on scale.

The Cantor set, Sierpinski carpet, and Menger sponge are older examples of fractal geometry that were constructed with a similar approach. They are all worth analyzing mathematically, but they have also been used to solve numerous problems that never crossed my mind before. For example, Mandelbrot associated the Cantor set with the distribution of faulty data transfer within wires. The Cantor set is a fractal constructed by dividing a line into three equal parts and then removing the middle third. This is repeated by doing the same thing to the remaining line segments until they look like tiny dots. The Menger sponge is a three-dimensional extension of the Cantor set where smaller cubes are removed from a larger cube. This is then repeated by dividing the remaining structure into smaller cubes and removing even smaller cubes from them. The resulting structure, which has an infinite surface area yet encloses zero volume, was used for the first time in the Eiffel Tower.

Other scientists also pointed out numerous examples of this branch of mathematics before it gained its official name in the last quarter of the twentieth century. Fractal geometry can be seen in many things, such as the arrangement of a fern’s petals, the route of a lightning strike, the pattern formed by a fracture in glass, the structure of the bronchi in the lungs, and the structure of our vascular system. It possesses its own tiny examples in the first cloud of gas and dust that was the starting point of the universe. Now I understand this, yet I struggle to understand why we waited for such a long time to discover it.

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