Sports billiards produce precise patterns: 'It's become a bit of an obsession'

“When the students showed me the first simulations, we thought this couldn’t be true. So we tested it several times, with different codes, in different computer languages. But we kept seeing these shapes.”

Maziar Jalal, a physicist at the University of Amsterdam, and his students found this hard to believe. Their very simple simulation of “billiards with memory” produced detailed, ever-changing symmetrical patterns that are somewhat reminiscent of fractals or symmetrical Islamic artwork. she published its results In the journal Physics Physical review letters.

It started as a spontaneous idea during the pandemic, says Jalal, an Iranian-American, who ended up in Amsterdam via Canada, Twente and Cambridge. “I was stuck at home and reading a lot of Maryam Mirzakhani’s work.” Mirzakhani (Iran, 1977-2017) was the first female winner of the Fields Medal. I looked into “sport billiards,” a kind of idealized version of the green cloth found in a café.

A wealth of results

A sports billiard ball has no dimensions and no friction, so it keeps bouncing off the walls. Jalal: “For example, the ball can return to its path, or continue to explore new pieces of fabric.” Non-quad billiard tables were also explored, from triangular and circular to the very complex, with a wide range of mathematical results. Jalal: “I had the idea that you could use a pool table as a simple form of active material.”

Active matter, a currently popular branch of physics, is about the collective behavior of simple components that actively contribute to their movement: crawling insects or sperm cells moving with their tails. Jalal: Now suppose you gave him some form of memory. There are ants that secrete scent trails that other ants follow. In fact, slime molds avoid their own paths. In this way, a kind of memory is recorded in space.

Suppose the billiard ball not only bounces off the walls, but also bounces along the path it left behind previously. Sooner or later, the particle will encounter its own path. Jalal: “After a few hours of playing with the idea, I discovered that the molecules always get stuck in the end.” Sometimes that end came very quickly, and other times it took a long time. It was certainly difficult to predict the location of the accident.

“I tried to prove things mathematically, but to be honest, as a physicist I didn't achieve much,” he admits. In the Netherlands, together with master's students Theis Albers, Stijn Delenweg and Nico Schramma, he decided to take a physics-based approach: simply simulate the path of hundreds of millions of billiard balls in a computer. “This gives you a distribution of path lengths and a density map of where the particle gets stuck.”

And then this wonderful complexity emerged, which also showed chaotic behavior: a small difference in the starting point makes a big difference in the end point. This produces fine details in density maps. “You can enlarge these patterns and continue to find new patterns endlessly.”

Reminiscent of fractals

This is reminiscent of fractals, the famous self-similar structures that can be enlarged to infinity and that are the product of simple mathematical equations. “But I don’t think they are fractals, because they don’t look like themselves.”

The richness of forms raises further questions. “With polygons with an odd number of sides, the particles often end up in the middle, while with an even number of sides, they end up at the edge. No idea why.” In some density maps, dark spots indicate regions where the particle never ends. Most particles break down very quickly, but some last hundreds of times longer, again for reasons that are largely unclear. Jalal finds it particularly interesting that such a simple, basic idea leads to so much complexity. “It's become a bit of an obsession, not a day goes by that I'm not working on this or reading about it.”

Jalal believes that applications may lie in research on animal behavior or search for food, or in improving search strategies in rescue work. “And you can think about cryptographic algorithms, where unpredictability is an advantage.”

Although this post is basically a very preliminary exploration. “I hope this will inspire not only physicists, but also mathematicians. I actually think they can make better progress on this than we can.”

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