College College Physics and Mathematics TSS

Fractals: The Beauty of Repetition

From complex math problems to the coast of Britain, fractals show up everywhere.

From complex math problems to the coast of Britain, fractals show up everywhere. You may have briefly heard of them in your math classes; they’re complex never-ending patterns. A fractal is self-similar, so if you zoom into it, it appears the same as it did originally (1). For example, if you zoom into a picture and it looks the same as it did before you zoomed in, it’s a fractal. You can keep repeating this infinitely in a pure fractal and it will always look the same as the original. Pure mathematical fractals are infinite, but a fractal can be finite if it’s approximated (1).

A famous example of a fractal you can easily zoom in on is the Mandelbrot Set. Pictured below, this pattern of bubbles and colors continuously repeats itself at every level. A more mathematical looking example of a fractal is the Cantor Set, pictured below. The Cantor Set starts with a line. To begin the fractal, you cut out the middle third of the line and put the new line under that one. You continue to do this infinitely, and it always looks the same in the next level as it did before (2).

 

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Image from Research Gate

It’s cool to see these repeating patterns, but fractals are also extremely important. For some reason, they appear everywhere. You can find fractals in vegetables, coastlines, and even human DNA. In normal geometry, we can’t see the shapes that trees make or an accurate structure of mountains. Normal geometry can’t very accurately portray nature. Fractal geometry, however, does just that (3). By repeating simple formulas multiple times to create a fractal, you can begin to see shapes that resemble trees, river systems, and mountains with close accuracy (1).

Many things that are pleasing to the eye are fractals or resemble them. Our brain likes repetition, patterns, and neatness. Just like how the Golden Ratio is present in many aesthetically pleasing structures, fractals are extremely present in beautiful structures in nature. For example, humans might find seashells or snowflakes appealing because of their fractal properties. Fractals have also been found in the human genome; chromosomes repeat in certain patterns multiple times within DNA (4). Mathematicians still aren’t sure why fractals are so present all around us or why repeating a function in every iteration creates something a fractal every time (1), but we know that they’re important. In the meantime, though, they’re really cool to look at.

 

 

References:

(1) Dallas, George. “What are Fractals and why should I care?” WordPress, https://georgemdallas.wordpress.com/2014/05/02/what-are-fractals-and-why-should-i-care/. Accessed 12 August 2018.

 

(2) Barile, Margherita and Weisstein, Eric W. “Cantor Set.” MathWorld, http://mathworld.wolfram.com/CantorSet.html. Accessed 12 August 2018.
(3) “What are Fractals?” Fractal Foundation, n.d., https://fractalfoundation.org/resources/what-are-fractals/

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