So What?

We’ve seen an example of a startling one-dimensional fractal, but what can fractals be used for? Mandelbrot saw fractals as essential to understand nature, he famously said the following:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. — Mandelbrot

The cornerstone of fractals lies in the fact that they are similar at different scales. We can zoom into the Koch Snowflake and see the same pattern as we go further and further in. This feature is known as “self-similarity” and shows up all over in nature. Fractals show up in coastlines, trees, blood vessels, and of course snowflakes. For a deeper dive into mathematicals patterns in nature, I really like this page!

An interesting consquence of fractal theory is the resulting dimensions. It can be shown that the Koch Curve described above has a dimension of approximately 1.26! This is a tough concept to understand, how can we have a dimension that isn’t an integer? Dimension was first described in this way by Felix Hausdorff and described the “roughness” of an object. For a beginning explanation about this, and some more examples check out this page.