Leonard Kelley holds a bachelor's in physics with a minor in mathematics. He loves the academic world and strives to constantly explore it.
The father of fractals is Benoit Mandelbrot, a gifted mathematician who evaded the Nazis in his youth and later worked for IBM. While at IBM, he worked on a noise problem that telephone lines seem to have. It would stack up, accumulate, and ultimately destroy the message being sent. Mandelbrot wanted to find some mathematical model to find the properties of the noise. He looked at the bursts seen and noticed that when he manipulated the signal to change the noise, he found a pattern. It was as if the noise signal was replicated but at a smaller scale. The pattern reminded him of a Cantor Set, a construct of math that involved taking the middle third of a length out and repeating it for each subsequent length. In 1975, Mandelbrot coined the term "fractal" for this kind of pattern, but the term didn’t catch on in the academic world for some time. Ironically, Mandelbrot wrote several books on the topic and they have since become some of the bestselling math books of all time.
Properties of Fractals
Fractals have finite area but infinite perimeter because of the consequence of our change in x as we calculate those particulars for the given shape. Our fractals are not smooth curves (like a perfect circle) but instead are rugged, jagged, and full of different patterns that ultimately end up repeating no matter how far you zoom in and also cause our most basic Euclidean geometry to fail. But it gets worse: Euclidean geometry has dimensions that we can easily relate to, but now do not necessarily apply to fractals. Points are 0 D, a line is 1 D, and so on, but what would a fractal’s dimensions be? It seems like it has area but it is a manipulation of lines, something between 1 and 2 dimensions. Turns out, chaos theory has an answer in the form of a strange attractor, which can have unusual dimensions usually written as a decimal. That leftover portion tells us which behavior the fractal is closer to. Something with 1.2 D would be more line-like than area-like, while a 1.8 would be more area-like than line-like. When visualizing fractal dimensions, people use different colors to distinguish between the planes that are being graphed (Parker 130-1, 137-9; Rose).
Koch snowflakes, developed by Helge Koch in 1904, are generated with regular triangles. You start by removing the middle third of each side and replacing it with a new regular triangle whose sides are the length of the removed portion. Repeat for each subsequent triangle and you get a shape resembling a snowflake (Parker 136).
Sierpinski has two special fractals named after him. One is the Sierpinski Gasket, where we take a regular triangle and connect the midpoints to form 4 total regular triangles of equal area. Now leave the central triangle alone and perform again for the other triangles, leaving each new inner triangle alone. A Sierpinski Carpet is the same idea as the Gasket but with squares instead of regular triangles (137).
As is often in mathematics, some discoveries of a new field have prior work in the field which wasn’t recognized. Koch snowflakes were found decades before Mandelbrot’s work. Another example is Julia Sets, which were discovered in 1918 and were found to have some implications for fractals and chaos theory. They are equations involving the complex plane and complex numbers of the form a+bi. To generate our Julia Set, define z as a+bi then square it and add a complex constant c. Now we have z2+c. Again, square that and add a new complex constant, and so on and so forth. Determine what the infinite results for this are, and then find the difference between each finite step and the infinite one. This generates the Julia Set whose elements don’t have to be connected in order to form(Parker 142-5, Rose).
Of course, the most famous fractal set has to be the Mandelbrot Sets. They followed from his work in 1979 when he wanted to visualize his results. Using Julia Set techniques, he looked at those regions between finite and infinite results and got what looked like snowmen. And when you zoomed in at any particular point, you eventually got back to the same pattern. Later work showed other Mandelbrot Sets were possible and that Julia Sets were a mechanism for some of them (Parker 146-150, Rose).
Parker, Barry. Chaos in the Cosmos. Plenum Press, New York. 1996. Print. 130-9, 142-150.
Rose, Michael. “What Are Fractals?” theconversation.com. The Conservation, 11 Dec. 2012. Web. 22 Aug. 2018.
© 2019 Leonard Kelley