Lee Coursey, Flickr

The fractal nature of almost all things

Understanding nature’s fractals, from galaxies to ecosystems to the human heartbeat

There are no straight lines in nature.

Here’s a short activity: take a bowlful of dried rice, or, if your environment allows, sand. Pour it slowly onto the same spot. Watch as it builds into a pyramid. Continue to watch as the sides of that pyramid begin to avalanche. Notice how these avalanches continue to occur at the same frequency while you continue to pour.

This frequency is the frequency of something that is alive and growing, gaining matter, in a non-linear, self-repeating way. This is the frequency of a fractal.

Or, look at a tree, ideally, if your environment allows, when it is bare. Look at how the trunk leads up to a branch that splits off into more branches, then smaller branches, smaller branches, and finally branches so small that they appear scratches in the sky. This, again, is the self-repeating, slightly imperfect pattern of a fractal. (If your trees are leaved, look at the leaves, and you will notice the same patterns in the veins.)

The term fractal was coined by Polish-born mathematician Benoit Mandelbrot, rooted in the Latin word “fractus,” which means “broken glass,” the reference being to broken lines. In the 1980s while working at IBM, Mandelbrot arose as a singular pioneer into the fractal universe, publishing the seminal work The Fractal Geometry of Nature, which laid the mathematical and philosophical foundation for fractal theories that have since been adopted by analysts, scientists, and theoreticians from all disciplines – from art to astrophysics, economics to the natural sciences, social media to medicine, and climatology to human psychology.

The most accessible demonstration of Mandelbrot’s geometry is the Mandelbrot set, a visual self-repeating pattern that zooms in on itself to the infinite. It’s hypnotic and trippy, as soothing and ruminative as staring up at a clear night sky. Or at a lightening storm. Or waves and coastlines, cascading waterfalls, a view of endless mountains, a head of radicchio sliced in half, the distribution of galaxies, the lines and veins in the palm of your hand.

These are all the self-repeating, non-linear patterns of fractals.

The cylindrical fractals of the Epithelantha bokei species of cactus. Pavel Golubovskiy, Flickr
The cylindrical fractals of the Epithelantha bokei species of cactus. Pavel Golubovskiy, Flickr

Sights (un)seen

The thing about fractals is that once you get a sense of what they are, you’ll begin recognizing them everywhere. Stock market patterns? Fractals. The computer-generated landscapes in videogames and films? Designed with fractal geometry. The viral sharing of something on social media – or epidemics like the novel coronavirus? Spread with the branched repetition of fractals.

Before it gets too grand, let’s look at the self-repeating patterns in our own nature. American physicist Richard Taylor, who applies fractal understanding to the development of nanoelectronics, led a group of scientists in conducting extensive eye-tracking tests, finding that if the human eye is given an image it has never seen before, it will scan it with a fractal dimension of  D = 1.5. This means that it neither moves randomly across the image, nor moves across it from one side to the other in an organized way. It rather moves in a “mid-fractal trajectory,” in a pattern of small and long jumps, so as to cover the whole image in the most efficient manner.

A drip-painting of abstract expressionist artist Jackson Pollock. Dale Cruse, Flickr
A drip-painting of abstract expressionist artist Jackson Pollock. Dale Cruse, Flickr

Albatross and sharks, when circling over or in a body of water in search of a fish dinner, will cover the territory with a pattern of the same fractal dimension of 1.5. The island-like drips in Jackson Pollock’s abstract expressionist paintings, Taylor has found, also display a dimension of 1.5, and he’s subsequently found that people are more readily drawn to Pollock’s paintings than other pieces of abstract expressionism created with less organized construct in their chaos.

The dimension of 1.5 lands squarely in the middle of the fractal dimension range that falls between but is not equal to 1 and 2, with the latter indicating more complex fractals in two dimensions (in three dimensions, fractal dimensions fall between but are not equal to 2 and 3), like capillaries in lungs, neural networks and brainwave oscillations, or the slightly irregular repetition of the human heartbeat.

“That’s a little bit like the biophilia hypothesis where you can say, okay, the reason why we respond positively or feel affiliation to these fractal patterns is, perhaps, because we are also, ourselves, a fractal object in a sense,” says Swedish landscape architect and environmental psychologist Caroline Hägerhäll, who has worked closely with Taylor throughout her career and employs in her work a rigorous understanding of the interlinkages between fractal geometry and the restorative properties of nature.

The fractal patterns of a spiraled fern. fdecomite, Flickr
The fractal patterns of a spiraled fern. fdecomite, Flickr

“If we understand this more in detail, maybe we could design particular environments, with fractal sounds or visual fractals to help people with particular specific needs,” she says. Some current examples being researched and considered are hanging landscape or fractal-based art in hospital rooms, incorporating time in nature into the treatment of trauma patients, and sending images of fractals into space with astronauts, to give them a source of grounding and natural respite.

“Our inner fractals can resonate with the outer fractals, and we can have a kind of continuity and a sense of wholeness with the environment,” says Franco Orsucci, a leading clinical scientist and professor at the University College London who has focused his research on fractals since the 1980s’ Mandelbrot heyday.

Fractals have been a bedrock principle of Orsucci’s numerous journal articles, examining everything from the relationship between happiness and deep ecology to the stimulation of the brain’s alpha waves to enhance heart-rate variability.

Even his hobbies cannot escape the fractal grasp. “The distribution of olive trees is pretty fractal,” he says of the grove at his Tuscan home. “They self-organize pretty well and can produce pretty good quality olive oil when they decide.” He’s also a jazz guitarist, not least because “jazz music is pretty much fractal.”

Fractals are inherent to the structures of ice and snowflakes. Ivan Turkouvski, Flickr
Fractals are inherent to the structures of ice and snowflakes. Ivan Turkouvski, Flickr

While jazz or Mozart are often said to have fractal-like patterns in their compositions, however loosely so, Orsucci also speaks of a more stringently fractal set of sounds, known as pink noise. In terms of pitch structure, pink noise falls in between white noise (think TV static or homogenous buzzing) and brown noise (thunder or crashing waves), exhibiting a fractal correlation between pitches over time. Think softly falling rain, or gently rustling leaves.

“It’s kind of disordered, but in a quiet, nicely ordered way,” describes Orsucci. “It’s considered to be relaxing.”

Binaural beats, measured in hertz and often applied to mindfulness or meditation practices, as well as achieving states of deep focus in work, are based on the frequencies of pink noise. These frequencies are the same as those of the rice and sand avalanches, as well as the movements of the eye and albatross. “Obviously they tune our neurons to respond to synchronize to that kind of rhythm,” he says.

And therein lies the paradox

Perhaps you’re getting a better idea of fractals now. Maybe you’ve looked at a tree or thought about the ocean or taken a deep breath, imagining your lungs, and fractals are beginning to seem familiar.

Now is probably not, then, the most encouraging time to note that they never did make sense to Nobel Prize–winning quantum theorist Richard Feynman, whom The New York Times described in his obituary as “arguably the most brilliant, iconoclastic and influential of the postwar generation of theoretical physicists.”

“I was in a drawing class with the physicist Richard Feynman, and we were meeting weekly to draw together at a sculptor’s home,” says California-based clinical psychologist Terry Marks-Tarlow. “When I first came across a fractal, I immediately ran to Dick [Feynman] and said, ‘Don’t you think fractals are profound?’ And after giving a state of the art, technical definition of fractals to somebody else, much to my surprise, he turned to me, and he said, ‘I don’t understand them.’ ”

Perhaps this is because part of the fun of fractals is that the deeper into their patterns you go, the more they elude understanding.

Waves demonstrate the pattern frequencies of fractals. Harald Philipp, Flickr
Waves demonstrate the pattern frequencies of fractals. Harald Philipp, Flickr

The most common example of this is measuring a coastline. With more than 17,000 islands, the archipelago of Indonesia is said by the Central Intelligence Agency to have 54,716 kilometers of coastline. But think if you measured the circumferences of these islands not in kilometers but in meters, or decimeters, centimeters, millimeters, micrometers. The smaller the measurement unit, the longer the coast becomes, like watching the Mandelbrot set and realizing it will never end.

Thus, the coastline of Malta could be greater than that of the U.S. The sun could be farther away than Pluto. Lung capillaries longer than the roots of a redwood tree.

To return to the first point of this article, the reason why straight, easily measurable lines don’t exist in nature can be summarized in a single word: resiliency.

Anyone who has ever been coached in a sport has likely been made to stretch before or after practice, to keep muscles flexible and limber, less susceptible to sprains and strains. Similarly, ecosystems with more complex levels of biodiversity are healthier and more resilient to affliction and threat: monoculture forests and plantations are known to be far more vulnerable to diseases or the impacts of climate change than intact natural forests. A single tree is patterned so as to sway, not break, in the wind.

The repeating patterns of a snail's spiral. blese, Flickr
The repeating patterns of a snail’s spiral. blese, Flickr

So it goes, too, for a healthy heart. The heartbeats of many people with cardiovascular disease have been found to be overly regular, and in his aforementioned findings, Orsucci argues that the stimulation of certain brain waves can re-introduce more irregularity into the heartbeats of recovering patients to positive effect.

“Practically, fractal distribution or fractal shape is more naturally flexible,” says Orsucci. “The system is more flexible to respond to any environmental stressor or stimuli or any kind of interaction.”

Something personal

In her work as a psychotherapist, Marks-Tarlow applies fractal geometry to behavioral change as well.

“You can look at repetition compulsion as a fractal pattern, that people keep repeating the same patterns over and over again. In some people, that repetition is more linear in that it’s more exact, and that’s not very healthy. A healthier variety has a similar pattern, but it’s not exactly the same. It’s more complex, higher dimensional, with more variation. There’s room for evolution of the underlying pattern.”

In short: an understanding of how fractals and their growth patterns in nature lends itself well to healing and personal growth.

Glowing gas and dark dust in the Large Magellanic Cloud galaxy. NASA Goddard Space Flight Center
Glowing gas and dark dust in the Large Magellanic Cloud galaxy. NASA Goddard Space Flight Center

In the same vein, Orsucci says that many lives develop in a fractal-like manner, with one thing leading to another, in a self-similar but not always controllable manner. “It’s good to realize that sometimes a small intervention can produce a very big change,” he says, reflecting on the path his own life. “And sometimes, if you think you are the creator and you expect something big, it’s not going to happen. You have to understand the way… the way to help a system or a person self-organize in the best way for themselves.”

Marks-Tarlow extends her research to the metaphorical applications of fractals, such as in certain spiritual practices such as Buddhism that seek to teach the interconnectedness of all things. “We’re in the universe at the very same time that the universe is in us, and that’s absolutely profound,” she says.

In patterned form, ending near to where this all began, consider the words of Ryōkan, one of the greatest Zen Buddhist poets of the Edo period:

Too lazy to be ambitious,
I let the world take care of itself.
Ten days’ worth of rice in my bag;
a bundle of twigs by the fireplace.
Why chatter about delusion and enlightenment?
Listening to the night rain on my roof,
I sit comfortably, with both legs stretched out.

Article tags

artbeautycoastal ecosystemsforestsfractalfractalshappinesshuman wellbeingInternational Day of HappinessMandelbrotnaturepatternsSustainable lifestyletreeswellness



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