An image from newly published research paper "Jewels from chaos: A fascinating journey from abstract forms to physical objects."
Francesca Bertacchini, Pietro S. Pantano, Eleonora Bilotta

What Is the Butterfly Effect? How Scientists Find Beauty in Mathematical Chaos

Researchers are using chaos theory to 3D-print stunning pieces of jewelry. As Einstein himself said, "The greatest scientists are always artists as well."

As a hard-working meteorologist, Edward Lorenz wanted to predict the future. As an astute mathematician, he quickly learned he could not. 

In the 1960s, during the peak of his weather-forecasting career, Lorenz plugged a bunch of atmospheric coordinates into a computer program that simulated weather patterns. The goal was to learn what conditions humanity would be in store for over the next few months. Easy. Straightforward. And sure enough, Lorenz got some answers. But then, like all good scientists, he decided to run the program a second time. Just in case. 

While that happened, Lorenz went to pour himself a cup of coffee in the lobby of his lab. When he returned, he was astonished. 

Every prediction, despite coming from the same atmospheric inputs, was totally new on run No. 2. 

"The numbers coming out of the printer had nothing to do with the previous ones," Lorenz wrote in a book about his experience years later. After some fiddling, his surprise only grew deeper. 

Turns out, the computer had rounded his inputs an eensy bit differently during its second run, and I mean infinitesimally eensy. Yet such minuscule adjustments single-handedly changed the future -- to a drastic degree. It was almost like a butterfly flapping its wings one day could start a chain reaction leading to a hurricane on the other side of the world the next. 

Eventually, Lorenz came up with a name for his weather mishap that seemed to place randomness atop a table of underlying order: the butterfly effect

And the butterfly effect would soon blossom into an entire, brilliant field of study known as chaos theory. It was thrilling to realize that some futures, like the weather, can scientifically defy the rules of determinism, that they can remain unknown to us until we physically live through them.

Where there is imperfection, there is beauty

Fascinatingly, the unpredictable nature of weather that Lorenz discovered some 60 years ago opened the floodgates for mathematicians, philosophers, physicists, artists -- and now even for designers of jewelry, according to a newly published research paper. But first, let's explore how we've reached this point. 

A serendipitous thing about chaotic systems is that when you graph their movements, they look just as striking as you'd expect. At risk of simplification, sets of data that result these sort of cool mathematical shapes are known as strange, or chaotic, attractors

A swirling golden graph on top of a black background.

A plot of the Lorenz attractor.

Wikimol, Dschwen via Wikipedia

So of course, chaos theory started a race among scientists to understand what happens when a system moves from a point of stability to a mess of infinite instability. A classic example of this is with a double pendulum. A normal pendulum (think, grandfather clock) has a pretty peaceful path. Left, right, left, right. No major speed changes. But add a second pendulum to the end of the first and you'll find the double pendulum's path gets all wonky.

This weird pendulum's movements will never be predictable again because of its immense sensitivity to the system's initial conditions. To predict where it's going to go, you'd need to know its starting point with 100% certainty. That's simply impossible. Voila, you've made a chaotic pendulum.

Some scientists are also interested in decoding what's known as the "edge of chaos," which denotes the tipping point between chaos and its counterpart. Potentially, tangles of neurons in our brains live right on this stressful line, which means that understanding their inner workings could absolutely revolutionize neurological treatment. 

But chaos theory quickly caught the attention of visual artists, sculptors and musicians too. People who search for beauty in imperfection and dissonance -- precisely the kinds of patterns that chaotic systems leave behind, including the disturbed pendulum's lawless path.

Eleonora Bilotta, an expert in chaos theory at the University of Calabria in Italy, is a scientist and artist who sees both sides at once.

"Our research group has been studying chaos theory for over 20 years, and during that time, we have made a major breakthrough by discovering more than a thousand chaotic attractors, starting with the Chua circuit," she said. 

More recently, however, her team has been working on translating the dynamics of these stunning systems into visual forms. 

"We created a bridge between the abstract world of mathematics and the more intuitive world of art and perception," she said.

Chua circuits, first discovered in 1983 by Leon O. Chua, are typically found in electronic circuits. And as Bilotta explains, they are often used in studies of chaos theory to help us understand how these systems work and translate to other fields like chemistry, physics and biology. And like with Lorenz's weather fluctuations, small changes to the Chua circuit's parameters have the potential to lead to massive changes in the system's behavior. 

It's similar to the butterfly effect but is formally referred to as "bifurcation" in this case.

"One of the unique properties of the Chua circuit is that it has the ability to generate a wide range of chaotic attractors, each with its own distinct shape and characteristics," she said. 

This is a big deal, through an artist's eye. It means graphing the dynamics of Chua circuits specifically yields a flurry of exquisite patterns, and as Bilotta notes, it particularly creates designs associated with what are known as "fractal structures." 

A fractal sequence basically means an object's structure continuously breaks out into smaller and smaller versions of itself. You'll find these patterns in snowflakes, star explosions, trees -- even in your own body. 

With this in mind, Bilotta and her colleagues have already turned the Chua circuit's chaotic patterns into sound. 

"Music is a universal language that can be understood by people regardless of their background, and it can convey complex ideas in a way that is easy to grasp," she said.

But now, according to a research paper co-authored by Bilotta and published in late January in the journal Chaos: An Interdisciplinary Journal of Nonlinear Science, this butterfly effect artform has also found its way into jewelry. 

"Jewelry is a highly personal and wearable form of art, which allows people to connect with the chaotic attractors in a more intimate and personal way," she said. "We believe that there is a symbiotic relationship between art and science, where each can inform and inspire the other."

Finding a mathematical goldsmith

Mathematical diagrams looking at paramaters of chaotic attractors yet to be turned into jewelry.
Francesca Bertacchini, Pietro S. Pantano, Eleonora Bilotta

In 2007, Bilotta attempted to reproduce the squiggly designs of chaotic attractors in a way that most jewelry makers would approve of: finding a goldsmith. 

"But the results were not precise," she said. "While it is true that some artists have attempted to interpret these shapes in the past, the intricate fractal structures of chaotic attractors make them very difficult to reproduce by hand precisely." 

Traditional goldsmithing techniques couldn't smooth out the jewelry enough, she explained, or prevent holes in the shapes.

Various pieces of jewelry based on the team's chaotic attractors.
Enlarge Image
Various pieces of jewelry based on the team's chaotic attractors.

A selection of the team's chaotic attractors in the form of pure chaotic shapes. These jewelry pieces were molded using bronze and silver.

Francesca Bertacchini, Pietro S. Pantano, Eleonora Bilotta

The next step was to find a technique of generating chaotic jewelry that could handle the immensely detailed geometric figures a Chua circuit exudes. Bingo. 3D-printing. Or more specifically, resin 3D printing. But the goldsmiths' creative expertise didn't fade from the picture.

"We had to work closely with the goldsmiths, experimenting with different techniques and adjusting the digital designs in order to achieve a smooth and polished final product," Bilotta said. "It was a challenging process, but eventually, we were able to overcome these problems and create beautiful jewelry that accurately represented the chaotic attractors."

"We also plan to integrate artificial intelligence algorithms to further push the boundaries of chaotic design and discover new and unexpected forms and applications," she said, mentioning that her team wants to explore materializing other mathematical forms as well, such as what are known as "anti-Pythagorean" systems. 

The scientists and the interpreters

When I was in college, I had a friend in physics class who had studied ballet most of her life. Somehow, she managed to convince the department to allow her a mashed-up major in both the graceful dance and hard science typically associated with topics like solar system dynamics, electrical engineering and magnetism. 

I can only imagine it was because ballet is so clearly rooted in math and physics. There is a torque to every pirouette just as there is one to every complex satellite maneuver. In fact, every dance form and sport you can think of is too. Bill James, for instance, famously figured out how statistics underlies baseball theory, and I must offer a shout out to 2005's Ice Princess, a cute little film based on blending math and figure skating.

Six covers of the International Journal of Bifurcation and Chaos, each featuring the team's work.

To bring a vision of having chaotic jewelry in a museum (i.e., first image), the team created a 3D navigable application called the Museum of Chaos, which allows visitors to explore and interact with chaotic attractors in an immersive way (i.e., the last image).

Francesca Bertacchini, Pietro S. Pantano, Eleonora Bilotta

On the flip side, thinking about scientific theories like Albert Einstein's general relativity, marine biology and orbital chemistry pretty much requires a visual imagination. 

Neri Oxman, for instance, is a designer whose work based on the natural intricacies of crustacean shells and human breath was featured in New York's Museum of Modern Art in 2020, where Bilotta hopes to showcase her work one day.

"While art can help to make science more understandable and relevant, science can also provide new tools and inspiration for artists," Bilotta said, noting that "together, they can offer new perspectives and insights on the world around us, and help to deepen our understanding and appreciation of both art and science."

Various pieces of jewelry made by Bilotta and her team.
Francesca Bertacchini, Pietro S. Pantano, Eleonora Bilotta

Einstein himself once said that "after a certain high level of technical skill is achieved, science and art tend to coalesce in esthetics, plasticity, and form. The greatest scientists are always artists as well."

When Lorenz discovered the butterfly effect, he was excited to tell the world about a new fundamental, mathematical tenet of our universe. But when he came up with its name, he wasn't thinking about science. He was going for poetry.

This reminds me of a sentiment I once heard somewhere. Science is our medium of finding truth, and art is our means of interpreting it.