Interestingly, the idea that beaches could have features at multiple size scales played a huge role in the early development of fractal theory. "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension" by Mandelbrot (of the eponymous set) was, in 1967, one of the first papers to discuss the basics of fractals, although the term "fractal" itself was not coined until 1975.

In short, it discusses the impossibility of measuring the length of the coastline of Britain. Were you to measure it from a map, you might miss small sandbars or other features. Were you to walk it with a tape measure, you would still not get the edge of every rock. And even if you did, counting it one centimeter at a time, you would not trace the edge of each grain of sand.

Therefore, a different metric is needed, so that in addition to a crude length, a measure of the "roughness," and more importantly the self-similarity of that roughness, is needed. Enter the Hausdorff Dimension, the simplest construction of fractal dimension.

In sort, imagine the coastline of Britain as a lineform. Is it one dimensional or two dimensional? Normally, we'd say it's one dimensional, but if we look closely at it, it wiggles so much -- so infinitely much -- that it appears to fill space. But it's clearly not two dimensional, so we must construct a concept of noninteger dimension. The more self-similar the roughness, and the more the line appears to fill 2-space, the closer it is to being one-dimensional.

South Africa, with a very smooth coastline, has a Hausdorff dimension of 1.02, whereas west Britain has a dimension of 1.25, similar to that of a Koch Snowflake.

Anyway, this is a cool gif. Math is also cool.