About Fractals

Image Source: http://mathworld.wolfram.com/Fractal.html

What is a fractal? A fractal is a never-ending pattern that repeats itself on all scales. Fractals are formed by repeating the same process over and over again. No matter how much you “zoom in” or “zoom out,” the pattern looks the same. This is called “self-similarity” – no matter where you look, the figure is similar to itself.

 

Fractals can be found in nature, such as in the pattern of the leaves of a fern or a branches of a tree, and they can be generated geometrically and algebraically. Geometric fractals are created by repeating the same pattern of shapes over and over, like the Sierpinski triangle. The fractals shown above are well-known geometric fractals. The top row shows the Gosper Island fractal, the next row is the Koch Snowflake, the next row is the Box fractal, and the bottom row is the Sierpinski Triangle.

The Sierpinski Triangle is the fractal that is the focus of the Trianglethon. To see a continuously zooming video of a Sierpinski Triangle, visit https://www.youtube.com/watch?v=_S8RGh0EyV4&feature=youtu.be.

Algebraic fractals are generated by repeating certain algebraic formulas over and over and plotting the results on a graph. The Mandelbrot Set, shown below, is an example of an algebraic fractal.

To learn more about the algebra behind the Mandelbrot set, visit https://www.youtube.com/watch?v=NGMRB4O922I.

Why Fractals Are So Soothing

When Richard Taylor was 10 years old in the early 1970s in England, he chanced upon a catalogue of Jackson Pollock paintings. He was mesmerized, or perhaps a better word is Pollockized. Franz Mesmer, the crackpot 18th-century physician, posited the existence of animal magnetism between inanimate and animate objects. Pollock’s abstractions also seemed to elicit a certain mental state in the viewer. Now a physicist at the University of Oregon, Taylor thinks he has figured out what was so special about those Pollocks, and the answer has deep implications for human happiness.