Fractal geometry is a cult classic. The concept of a fractal is beautiful yet mysterious and likely one of the most popular math ideas out there. It's also often used to understand and interpret natural phenomena—an application caused by, but also contributing to, its popularity.
In truth, geometry is not the most fitting frame for natural things. Geometry is the study of objects, and objects are purely about size and shape. The natural world has additional properties such as color, weight, temperature, and many others. Still, it is possible to reduce this full picture into a question of geometry, and it does simplify things. Clouds, for example, span a continuous spectrum of opacity, but it's more common to think of them as objects. Even in research, we tend to focus on their geometry: their size and shape.
Cloud geometries are interesting and counterintuitive because they are fractal. I'll describe what this means through the lens of a study our research group recently submitted for peer review (link here).
Standard mathematical fractals are created by specifying some basic pattern and then repeating that pattern at smaller and smaller scales. That is what allows for trippy infinite-zoom videos:
The basic reason that fractals are visually compelling is the fact that they have structure at many different scales. In the above image, there is a large-scale structure, and when we zoom in, we find there is smaller-scale structure as well. An exact copy of the large scale, in this case.
Any fractal must have at least some structure at all scales. This is what it means to be a fractal. To be clear, here is an example of something that is not a fractal, a sine wave:
Here, there is a wave structure at large scales, but when we zoom in, that structure disappears. This is not a fractal because the structure only appears at one zoom level, or in other words, at one scale.
Repeating patterns like the Koch fractal above are generally not observed in nature, and certainly not in clouds. So what do I mean when I said that clouds are fractal? Well, it’s not any particular pattern that is important. It’s this property of having structure at all scales. Some fractals will have a lot of structure, others have a small amount of structure, but what’s important is that it is distributed across all scales.
The amount of structure is quantified mathematically using the "fractal dimension". More structure implies a larger fractal dimension. It's perhaps easiest to understand what this means by looking at examples—here’s a version of the Koch fractal that has variable fractal dimension:
At lower values, the sides become smoother and smoother, eventually becoming a one-dimensional line. At higher dimensions, the shape becomes a tightly packed mess of lines, eventually becoming so tightly packed that the object fills the entire (two-dimensional) triangular regions.
This variation is often described by saying objects with higher fractal dimension are "rougher", "more wiggly", or "more complex". You can see what people mean by that above. But really, I think a more accurate mental image is how “tightly packed” or “dense” the fractal line is at various scales. Partly, the reason is that the “density” framing defines the fractal dimension mathematically:
The only remaining question is how to measure object density and scale considered. There are many ways, but for now just think of "object density" as measuring how tightly packed the fractal line is for a given scale considered: for example, in the above animation, you could ask how many pixels are white (object density) over the whole image (scale considered).
This framing is far more useful than repeating patterns. Repeating patterns are mathematical idealizations with little connection to reality. “Structure at all scales”, however, is widely found in natural objects—for example, clouds:
These are from three different atmospheric simulations which produce clouds with fractal dimensions ranging from 1.3 (left), to 1.7 (center), to 2 (right). As you can see, the amount of structure also increases left to right.
Another key difference between natural fractals like clouds and mathematical fractals is that nature cannot have structure over literally all scales, just some wide range. You can't look too closely at a cloud without seeing individual water droplets or even molecules.
This becomes particularly challenging if you want to perform a fractal analysis using real data like, for example, a satellite image or the simulation output above. Any image is pixelated at small scales and only covers some limited area overall. These are both features of the image—not the clouds themselves—which is not what we care about. Take this image, for example:
Zooming way in (left), the image is pixelated and unrealistic. Zooming way out (right) all we see is a small square: again, unrealistic. So in practice we have to focus on intermediate scales between the size of the pixels and the size of the image (center). This problem becomes surprisingly thorny, and we had to write two whole papers to cover all the subtleties we encountered.1
Our more recent paper also considers the question we sidestepped above in detail: how to define "object density" and "scale considered". I'll show two visualizations of how this actually works in practice. In both cases, we are measuring the fractal dimension of cloud edge, so the density measures focus on cloud edge.
One method represents "scale" by circles of various sizes (red) and "object density" by the number of cloud edge pixels (also red):
We argue this method is probably best because it doesn’t require as much data for accurate results. We compared it to an older, more common method, which represents "scale" by overlaying a mesh of different sizes and "object density" by the number of mesh boxes that cover a cloud edge point (red):
Both methods are ways to operationalize the “object density” vs. “scale considered” distinction. But the details are not as important as the foundational idea: structure at all scales. All of the above discussion, and the discussion in our paper, was specifically about the structure of cloud boundaries. This is mainly because our group is interested in cloud edge for theoretical reasons.
But more broadly, “structure at all scales” is a fundamental property of the atmosphere.2 Almost every measurable atmospheric quantity has structure across a wide range of scales. This enables a profound opportunity to unify our understanding: it implies the wind in your face follows the same ruleset as the largest hurricane. It’s not fully understood why. I hope we find out someday.
Every cloud is different, but somehow, at a more basic level, every cloud is the same:
1In addition to the one being covered here, this one.
2The term is actually “scale invariance”.