Significance and Applications
Now that we've explored the background and the essentials of what fractals are, let's discuss some of the things that fractals can be used for.
Fractals have helped in the development of more realistic animation. Using fractals and triangulation, realistic mountain ranges can be generated using a computer. In one of the early Star Trek films, entire planets and solar systems were computer generated using the ideals of fractals. Further, in Star Wars episode III, animators used fractals to create more realistic lava flowing and spraying into the air. The video clips below shows a few of these examples.
Mountains and Star Trek: PBS Nova - Fractals cropped via ytCropper
Star Wars: PBS Nova - Fractals cropped via ytCropper
Jhane Barnes, a fashion designer, collaborated with a mathematician and physicists to sue fractals to create more interesting clothing pieces. She also uses fractals in her work designing carpets and other textiles. This shows that fractals have more than just a technical application. People around the world are intrigued by the shapes and patterns of fractals, and Barnes reflected that in the clothing she created.
Jhane Barnes: PBS Nova - Fractals cropped via ytCropper
Another application of fractals came about in 1988 by Nathan Cohen. Cohen was a HAM radio enthusiast but was living in an apartment complex where his landlord didn't want him to have a large antenna outside the building. Cohen developed an antenna that is built in a fractal shape that is able to pick up many signals with differing wavelengths. This type of antenna has enabled modern cell phones to pick up a cell signal, wifi, bluetooth, and radio without having a bunch of bulky antenna sticking out. This has helped the progression of wireless communication. The video below continues the details of this story. If you want to learn more about Cohen's invention and subsequent business, click here to visit his website.
Fractal Antenna: PBS Nova - Fractals cropped via ytCropper
Additionally, fractals can be used in determining the health of a person. A healthy heart beat will have a fractal structure. Further fractal geometry could be used in tumor and cancer detection. Typical blood vessels will have a fractal, bifurcation-like structure. In areas where tumors are developing, however, the structure of the vessels will be much more chaotic. While ultrasounds and other cancer detection technologies may not be able to create clear enough images of tumors, they may be able to detect abnormalities in blood flow patterns.
Further, biologists can use fractal geometry to determine the number of branches and leaves an individual tree has. After computing the CO2 intake of a single leaf, they can use their calculations to determine the CO2 intake of the tree. Interestingly enough, the fractal distribution of branch sizes in a tree is similar to the fractal distribution of the sizes of trees in a forest. From there, biologists can determine the CO2 intake of a forest to determine the impact that a forest has on the amount of CO2 in the atmosphere. This use of fractal geometry gives biologists and ecologists the necessary knowledge to help protect the environment.
At the end of the day, we have to ask, why do fractals matter to us? Often, fractals can give a meaningful and insightful explanation of chaos and events happening around us. There are patterns and iterations and scalings in nature's course -- whether in snowflakes, clouds, earthquakes, the weather, or economics. Additionally, fractal geometry can bring things to the table that other fields of mathematics cannot. Classical geometry is good at describing man-made creations, but fractals are good at describing nature and its patterns. In the PBS Nova film, it says that "fractal geometry shows the underlying order and structure of nature" (Nova). We can translate what we see in the natural world using the language of mathematics, particularly fractals. The value of fractal geometry comes from being able to describe and model the world around us with greater clarity and explanation.
Mandelbrot Quote -- Image Source: Quote Fancy
Check out the references and resources page for additional sources and places to learn more about the interesting field of fractal geometry.