The Butterfly Effect: How Chaos Theory Blurs the Line Between Math and Art

The Butterfly Effect: How Chaos Theory Blurs the Line Between Math and Art

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As an avid meteorologist, Edward Lorenz wanted to predict the future. As an astute mathematician, he soon realized he couldn’t.

In the 1960s, at the height of his career as a weather forecaster, Lorenz plugged a set of atmospheric coordinates into a computer program that simulated weather conditions. The goal was to find out what conditions humanity would face in the coming months. Merely. Merely. And indeed, Lorenz got answers. But then, like all good scientists, he decided to run the program a second time. In case.

Meanwhile, Lorenz went to the hall of his lab to pour himself a cup of coffee. When he came back, he was amazed.

Each prediction was entirely new for run #2, despite coming from the same atmospheric inputs.

“The numbers that came out of the printer had nothing to do with the previous ones,” Lorenz wrote years later in a book about his experiences. After some trial and error, his surprise only increased.

It turns out that the computer had rounded its inputs slightly differently on its second pass, and I mean infinitely slightly. But such small adjustments, on their own, changed the future on a drastic scale. It was almost as if a butterfly flapping its wings one day could set off a chain reaction that would lead to a hurricane halfway around the world the next.

Eventually, Lorenz came up with a name for his weather disaster that seemed to put luck on an underlying order table: the butterfly effect.

And the butterfly effect would soon evolve into a brilliant field of study known as chaos theory. It was exciting to realize that certain futures, like the weather, can scientifically defy the rules of determinism, that they can remain unknown to us until we experience them physically.

Where there is imperfection, there is beauty

Curiously, the unpredictable nature of the weather, which Lorenz discovered about 60 years ago, has opened the floodgates for mathematicians, philosophers, physicists, artists — and now even jewelry designers, according to newly published research. But first, let’s take a look at how we got here.

A happy coincidence with chaotic systems is that when you plot their movements, they look as flashy as you’d expect. At the risk of oversimplifying, data sets that produce these kinds of cool mathematical shapes are called weird or chaotic attractors.

A representation of the Lorenz attractor.

Wikimol, Dschwen via Wikipedia

So, naturally, chaos theory sparked a race among scientists to figure out what happens when a system goes from a point of stability into a tangle of infinite instability. A classic example of this is a double pendulum. A normal pendulum (think grandfather clock) has a fairly smooth course. Left, right, left, right. No big gear changes. But add a second pendulum at the end of the first and you will find that the trajectory of the double pendulum becomes quite wobbly.

The movements of this strange pendulum will never again be predictable due to its immense sensitivity to the initial conditions of the system. To predict where it will go, you would need to know its starting point with 100% certainty. It’s just not possible. Well, you made a chaotic pendulum.

Some researchers are also interested in deciphering the so-called “chaos edge”, which denotes the turn between chaos and its counterpart. Potentially, tangles of neurons in our brains live along this stress line, meaning understanding their inner workings could revolutionize neurological treatment.

But chaos theory also quickly attracted the attention of visual artists, sculptors and musicians. People who seek beauty in imperfection and dissonance – the very kind of patterns that chaotic systems leave behind, including the anarchic path of the disturbed pendulum.

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

“Our research group has been studying chaos theory for more than 20 years, and during this time, we have made a major breakthrough by discovering more than a thousand chaotic attractors, starting with the Chua cycle”, a- she declared.

More recently, however, his team has worked to translate 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 by Leon O. Chua in 1983, are commonly found in electronic circuits. And as Bilotta explains, they’re often used in chaos theory studies to help us understand how these systems work and extrapolate them to other fields like chemistry, physics, and biology. And as with Lorenz weather fluctuations, small changes in Chua circuit parameters can cause it firmly System behavior changes.

This is similar to the butterfly effect, but in this case it is formally called a “bifurcation”.

“One of the unique properties of the Chua circuit is that it is able to generate a wide range of chaotic attractors, each with its own shape and properties,” she said.

It’s a big deal, through the eye of an artist. This means that the graphical representation of circuit dynamics Chua specifically produces a barrage of exquisite patterns, and as Bilotta notes, designs associated with so-called “fractal structures” are particularly created.

Basically, a fractal sequence means that the structure of an object continually breaks down into smaller and smaller versions of itself. You can find these patterns in snowflakes, stars, trees – even in your own body.

With that in mind, Bilotta and his colleagues have already transformed the chaotic patterns of the Chua circuit into sound.

“Music is a universal language that can be understood by people regardless of background, and it can convey complex ideas in an easy-to-understand way,” she said.

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

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

Find a math expert

Francesca Bertacchini, Pietro S. Pantano, Eleonora Bilotta

In 2007, Bilotta attempted to replicate the wavy designs of messy attractors in a way most jewelry makers would approve of: finding a goldsmith.

“But the results weren’t accurate,” she said. “While some artists have attempted to interpret these shapes in the past, the complex fractal structures of chaotic attractors make them very difficult to accurately reproduce by hand.”

Traditional silversmithing techniques cannot sufficiently smooth jewelry, she explained, or prevent holes in molds.

Enlarge image

A selection of the team’s chaotic attractors as pure chaotic forms. These jewels were made of bronze and silver.

Francesca Bertacchini, Pietro S. Pantano, Eleonora Bilotta

The next step was to find a technique for creating chaotic jewelry that could handle the immensely detailed geometric figures that a chua cycle exudes. Bingo. 3D printing. Or more precisely, resin 3D printing. But the creative skill of goldsmiths has not disappeared from the picture.

“We had to work closely with the goldsmiths, experimenting with different techniques and adjusting the digital designs to achieve a smooth and polished end product,” Bilotta said. “It was a difficult process, but we were ultimately able to overcome these issues and create beautiful jewelry that accurately represents 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 also wants to explore materialization. other mathematical forms, such as z “anti-Pythagorean systems”.

Scientists and interpreters

When I was in college, I had a friend in physics class who had studied ballet for most of her life. Somehow, she managed to persuade the faculty to allow her a combined major in graceful dancing and hard science, usually associated with subjects like solar system dynamics, electrical engineering, and magnetism.

I can only imagine it was because ballet is so clearly rooted in math and physics. Every pirouette has a torque, just like every complex satellite maneuver. In fact, it’s every dance form and every sport you can imagine. Bill James, for example, discovered how statistics underlie baseball theory, and I must commend 2005’s Ice Princess, a neat little film based on a mix of math and figure skating.

To bring the vision of chaotic jewels into a museum (i.e. the first image), the team created a navigable 3D app called Museum of Chaos that allows visitors to explore and interact with attractors chaotic in an immersive way (i.e. the last frame). Picture).

Francesca Bertacchini, Pietro S. Pantano, Eleonora Bilotta

On the other hand, thinking about scientific theories like Albert Einstein’s theory of general relativity, marine biology, and orbital chemistry requires a bit of visual imagination.

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

“While art can help make science more understandable and relevant, science can also provide artists with new tools and inspiration,” Bilotta said, noting that “together they offer new insights and perspectives. about the world around us, and to deepen it”. can contribute to our understanding and appreciation of art and science.

Francesca Bertacchini, Pietro S. Pantano, Eleonora Bilotta

Einstein himself once said, “After reaching a certain level of technology, science and art tend to merge in aesthetics, plasticity and form. The greatest scientists are always also artists.

When Lorenz discovered the butterfly effect, he was thrilled to tell the world about a new mathematical principle in our universe. But when he came up with the name, he wasn’t thinking about science. He wanted poetry.

It reminds me of a phrase I heard somewhere. Science is our medium for finding truth and art is our medium for interpreting it.


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