Fractals: A Case Study in the Grand Canyon

“When you’ve marveled at how the erosion on a bank of a stream looks just like a scaled-down version of the erosion you’ve seen on a larger river, or that it even looks like a mini-version of the Grand Canyon, you weren’t fantasizing, it actually is.” 

— Geoffrey West, “Scale”

If you’ve ever had trouble wrapping your brain around the concept of fractals, you’re in good company. Famed quantum mechanics physicist Richard Feynman once admitted to not fully understanding them (in the same breath that he provided their precise mathematical definition). When I am standing in front of a scene like this, a cliff-side shot in the middle of a blizzard of the South Rim of the Grand Canyon, I’m less struck by the strange melange of desert red rock and white powdery snow and more drawn into the dizzying distribution of sediment, storm clouds, snow, and spruce. These fractal patterns are a familiar sight to us all: a self-repeating kaleidoscopic sight, with just enough discordance to make it completely enrapturing.

But you don’t need to trek all the way out to Arizona to experience them. And they are more than a vocabulary for explaining patterns in the everyday natural world.

The nature of fractals in life and art

Fractals have been generalized and applied to any number of domains that affect us — the banal to the extreme — including architecture, jazz, social media, art fraud, disease transmission, medical biomarkers, astrophysics, the stock market, and even the special effects landscapes you see in video games or in fantasy backdrops of Lord of the Rings.

Shockingly, it took scientists until the 1970s (more than 2,000 years from Aristotle’s founding of the study of formal logic) to discover what may seem obvious whenever you look closely at, say, the cauliflower in your kitchen, or the dirt run-off on a roadway outside your house after a big storm.

We use many words to describe fractal forms, including “crinkly”, “discontinuous”, “self-similar”, “rough”. But explaining when and to what to apply these adjectives can be tricky, even though fractal features are hidden in plain sight all around us. As soon as you start recognizing these beautiful shapes and patterns, I guarantee that you’ll see them everywhere and wonder how something so obviously embodied in our surroundings could have been missed by the early natural philosophers.

For myself, I can’t help but think about Lewis Richardson and Benoit Mandelbrot as I marvel at the size and exquisite, infinitely complex, fractal structure of this cliffside photo of the South Rim of the Grand Canyon shot on the heels of a violent desert snowstorm.

So where do fractals come from?

Some examples of these features in natural phenomena:

• Forests
• Mountain ranges
• Lightning
• Vegetables
• Clouds
• Smoke
• Dried up riverbeds
• Salt marshes
• Coastal shorelines
• Waves crashing on a beach
• Plant & animal vascular systems

Despite their ubiquity in the surrounding world, it took a thoroughly abstract experiment by the polymath Lewis Richardson to lay the groundwork for their “discovery.”

The insight came from trying to calculate the likelihood of war between neighboring nation-states. Richardson proposed longer borders would mean more wars. To prove this, he set about what appeared to be a simple task: calculating border lengths.

Combing through the existing body of research, Richardson was shocked to discover there was no consensus between the data at all. While Mt Everest had been successfully mapped to within a few feet of accuracy, there were discrepancies between the borders of Spain and Portugal and Belgium and the Netherlands that were hundreds of kilometers apart!

This is where fractals get tricky. Because to understand them, you need to do some minor intellectual contortionism and accept the following statement.

Yes, the coastline of Britain is infinitely long.

The Euclidean world introduced to us early in our geometry textbooks, and espoused by those ancient philosophers so bound to the Platonic ideals of smoothness and elegance, points to the convergence of a well-defined length. That’s to say lengths are fixed. The finer the resolution, the more they converge to a fixed value and closer we get to the capital-T Truth.

Not so with the borders of nation-states (often defined by natural phenomena like mountain ranges and coastlines).

Unlike the length of our Euclidean city blocks and perfectly straight bedroom walls, Richardson found that the closer he zoomed, the longer the lengths became. For lines like borders and coastlines, measurements increase at a logarithmic scale the finer the resolution.

“In other words, the wiggles and squiggles at one scale are, on average, scaled versions of the wiggles and squiggles at another. So when you’ve marveled at how the erosion on a bank of a stream looks just like a scaled-down version of the erosion you’ve seen on a larger river, or that it even looks like a mini-version of the Grand Canyon, you weren’t fantasizing, it actually is…Most physical objects have no objective length…so why did it take more than two thousand years for people to recognize something so basic and which now seems almost obvious?

Very likely this has its origins in the duality that emerged as we gradually separated from a close connection to the natural world and became more and more distant from the forces of nature that have determined our biology. Once we invented language, learned how to take advantage of economies of scale, formed communities, and began making artifacts, we effectively changed the geometry of our daily world and its immediate surroundings. In designing and manufacturing human-engineered artifacts, whether primitive pots and tools or modern sophisticated automobiles, computers, and skyscrapers, we employed and aspired to the simplicity of straight lines, smooth curves, and smooth surfaces…In this new world of artifacts we inevitably became conditioned to seeing it through the lens of Euclidean geometry…blinding ourselves…to the seemingly messy, complex, convoluted world of the environment from which we had emerged.

This was mostly left to the imaginations of artists and writers….In this new world, a length is a length is a length, and that’s it. Not so in the immediate ‘natural’ world around us, which is highly complex and dominated by crinkles, wrinkles, and crenulations.”

— Geoffrey West, “Scale”

Richardson’s research initially languished in publication purgatory (unhelped by its arcane title and the journal’s relative obscurity) until Benoit Mandelbrot parlayed the work’s significance to the masses in his 1967 paper in the journal Science.

What fractals mean to me: the Grand Canyon and beyond

Fractals. Aren’t they profound? Quintessential disordered — but in an elegant, ordered kind of way. A structure that gives natural systems the kind of resiliency to which smooth, straight lines cannot compete. Fractal shapes are naturally more flexible, more adaptable to change. I see them everywhere. I am enamored by their exotic patterns. And I am inspired by their simultaneous simplicity and complexity.

This idea resonates with me on a personal level. In some ways, they seem to embody my own paradoxical impulse: the desire to understand and systemize the world around me — organize, sort, catalog it — while simultaneously hoping it can never be understood. I think I’ve become much better at acknowledging the implied self-defeat, and can appreciate the ambiguity and the tension. When I encounter these self-similar and repeating patterns in nature, I feel a sense of continuity and ‘oneness with my environment.’ Isn’t it nice when fuzzy problems can be embodied so wondrously in the physical world?

Some researchers have even argued that an understanding of fractals in nature can inform our own mental health and personal growth. Linear patterns in behavior correspond to compulsive behavioral disorders as well as signs of heart disease. Essentially, we are fractal organisms finding solace in our inner state embodied in our outer environment.

For more on this topic, I highly recommend Mandelbrot’s “The Fractal Geometry of Nature” and Geoffrey West’s “Scale”. But be careful, they may fundamentally reshape how you see the world around you.