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

Clouds are usually not spheres, mountains are usually not cones, coastlines are usually not circles, and bark just isn’t smooth, nor does lightning travel in a straight line. — Mandelbrot

The cornerstone of fractals lies within the indisputable fact that they’re similar at different scales. We are able to zoom into the Koch Snowflake and see the identical pattern as we go further and further in. This feature is often called “self-similarity” and shows up throughout in nature. Fractals show up in coastlines, trees, blood vessels, and in fact snowflakes. For a deeper dive into mathematicals patterns in nature, I actually like this page!

An interesting consquence of fractal theory is the resulting dimensions. It may be shown that the Koch Curve described above has a dimension of roughly 1.26! This can be a tough concept to know, how can we’ve a dimension that isn’t an integer? Dimension was first described in this fashion by Felix Hausdorff and described the “roughness” of an object. For a starting explanation about this, and a few more examples try this page.