Clouds Are Not Spheres: A Brief History of Fractals
Fractals: A Real-Life Application of Art Introduction From the trees outside your window to the blood vessels in your body, fractals exist all around you. Clouds, lightning, and coastlines – they are all fractals. Even we are composed of fractals. Fractals are a part of everything. For centuries, people have recognized nature’s tendency to repeat itself. Take Hokusa’s The Great Wave off Kanagawa for example. In the woodblock painting, you can see the repeating patterns at the tips of the giant wave as it crashes down on the boats. Although people noticed these patterns, they had not a clue what they were really looking at: a fractal.
Fractals: A Brief History Antennas, fractal compression, understanding the stock market, and designing more efficient computers—these are all possible because of our understanding of fractals. Before French-American mathematician Benoit Mandelbrot invented the branch of fractal mathematics, our understanding of how things function and why things repeat was limited. Without his discovery, we would not live in the world we live in today. Mandelbrot was contemplating algebra one day when began seeing vivid, geometric images in his mind. He soon realized that he had actually transformed algebraic formulas and equations into the pictures he was seeing. However, he kept this discovery to himself, for he figured it would not help him in the long run. Enter Georg Cantor, a Russian mathematician. In 1883, he created the first mathematical monster: He formed a straight line and broke it into thirds, erasing the middle portion. He repeated this process many a time, trying to fathom the mathematical mystery. He soon discovered that the length between the lines grew infinitely small and appeared to approach zero. However, the length never actually shrank to zero, and the pattern continued to repeat itself. Mandelbrot realized that these were types of fractals. He also saw different uses for them other than just “mathematical mysteries.” How Long is the Coastline of Britain? There was a long-standing issue in the cartography world: it was impossible to determine the length of a coastline. People knew that measuring a coastline with a yardstick for several miles would give them a rough estimate for its length; however, it would not be a very accurate measurement. They knew that the smaller the units of measurement, the more and more accurate the length would become. Mandelbrot took notice of this and wrote an article titled "How Long is the Coastline of Britain?". In it, he stated that coastlines were self-similar—that is, he could not measure the distance, but he could measure the roughness. This publication illustrated his first thoughts on fractals. Julia A short while later, Mandelbrot stumbled upon another monster: Julia. Julia was named after the French mathematician Gaston Julia who studied it during the First World War. Gaston Julia tried numerous times to iterate an equation in a feedback loop; however, each time he plugged a number into the formula, he would get a new number. Unable to create an image, he gave up; it was too much for one man to accomplish. Luckily for Mandelbrot, he had computers to aid him in his problem solving and was able to graph Julia using an IBM computer. It was a breathtaking image. Mandelbrot Set It was time for Mandlebrot to create his own equation. In 1980, he took all of the Julia sets and used them to create one iconic image. He named this image “The Mandelbrot Set.” Little did he know that it would become the most famous of all fractals. The Mandelbrot set and fractals were embraced by artists but not by mathematicians. His peers criticized and ridiculed him, claiming he was no good at math and that his fractals were merely nothing. His friends did not even speak to him. They called his fractals “artifacts from the computer” and called them utterly useless. Ignoring the responses from his colleagues, Mandelbrot created his soon-to-be-famous book, The Fractal Geometry of Nature. In it, he demonstrated how to measure things in nature and provided more ideas on the applications to his newly-discovered branch of mathematics. Soon afterwards, fractals were everywhere. Fractals Today In the early 1990s, fractal antennas were introduced. One of the very first of these was in the shape of the infamous “Koch snowflake.” It worked extremely well and had the advantage of being far smaller than a normal antenna. It could also receive more frequencies. The self-similarity of the fractal enabled it to work far better. Cell phone companies were also having similar issues at this time: every single portion of the phone ran at a different frequency. Luckily for them, the use of fractal antennas enabled them to all run on one single phone, thus enabling the cell phone revolution of today. Fractals are not only used in technology, but they are also used to understand the human body. For example: our heartbeats do not actually act like metronomes; instead, they fluctuate to several extremes, forming familiar patterns exactly like fractals do. These fractals also show up in blood vessels and have been used by doctors to help further comprehend tumors. Fractals are also used in movies. For example: one can create nature-like scenes such as mountains and even entire planets using only iteration. Believe it or not, the first film fractal came up in Star Trek II: The Wrath of Khan, where a planet was made entirely from a fractal. Even the last Star Wars movie had fractals. In one of the scenes, the protagonists and antagonists fight in a lava chamber. That lava was created purely with fractals. The artists added swirls to the 3-D model and shrunk them. They repeated this until the entire lava background was made of fractal swirls. As you can see, throughout several different fields, fractals have initiated revolutions of all kinds. We are fractals, and fractals are our future as well. Who knows where they will take us in the next twenty five years? We have gone from days where phones were the size of bricks to ones that can fit in the palm of your hand. We have even gone from days where film scenes were literally painted on film, now they're made on computers. Fractals have lead us this far, and who knows where they will lead us next?















