“My power flurries through the air into the ground.
My soul is spiraling in frozen fractals all around.
And one thought crystallizes like an icy blast-
I’m never going back; the past is in the past!”
– Elsa, Frozen
Before Frozen had even been released, people were raving about “Let It Go,” the song Queen Elsa sings as she declares her freedom to use her icy powers. I mean, you just knew this song was going to be big. It was going to win awards, it was going to be a karaoke staple, and it was going to drive parents crazy as their six-year-olds ran around the house singing it at the top of their lung.
It even manages to sneak in a reference to a term used in math, science, and art- “fractals.” But what is a fractal?
According to the Merriam-Webster dictionary, a fractal is “any of various extremely irregular curves or shapes for which any suitably chosen part is similar in shape to a given larger or smaller part when magnified or reduced to the same size.”
That’s quite a mouthful. Basically, if you have something with a type of pattern and you zoom in and see that same pattern on a smaller scale, that’s a fractal. This is referred to as self-similarity- a section of a whole looks very much like the whole itself. The small part is similar to the overall object.
Consider the below gif from the Fractal Foundation. We start with a triangle filled with other smaller triangles. As we zoom in on one of the inset triangles, we see that this pattern repeats, with smaller triangles making up larger triangles. Zooming in farther shows us the same pattern again… and again… and again.
Finite fractal features are everywhere in nature. This article on Wired shows many great and beautiful examples. One large bolt of lightning is made of many smaller bolts sparking out. Tree branches grow smaller tree branches, which grow even smaller twigs. Romanesco broccoli is composed of increasingly smaller domes forming spirals. Larger leaf veins feed into smaller veins.
And then, of course, there are frozen fractals. A snowflake forms when a water droplet in the atmosphere freezes. The tiny bit of ice begins to fall to the Earth’s surface. As it falls, more water latches on to the forming snowflake and freezes. Ice crystals build upon ice crystals, forming a fractal pattern. Repeating fractal patterns may also be seen on surfaces where water crystallizes.
Other than the cool images they produce, why are fractals useful?
Many models in all science fields are simple and can’t handle complex systems or beyond a certain level of variation. Fractal science and math considers the complexities– the parts with the parts within the parts to make up the whole. We might be able to only study a small part of something. If that part is representative of the whole, even if we cannot study the whole we still may be able to gain insight.
OK, now go listen to “Let It Go.” You know you want to.
Keep calm and science on.