Introducing a single rule to a theoretical game of billiards sparks a multitude of captivating mathematical inquiries and has practical implications in the physics of living creatures. Scientists are uncovering the captivating mechanics of billiards with memory. This week, a study conducted by the University of Amsterdam, with two master students as co-authors, was published in Physical Review Letters, delving into the enthralling dynamics of billiards with memory.
Billiards: A Mathematical Mystery
An idealized version of the game of billiards has been a source of fascination for mathematicians for many years. The fundamental question is a straightforward one: once a billiard ball is struck, where does it go and where does it ultimately end up? Assuming that the billiard table is flawless: the walls are perfectly elastic, there are no obstructions on the playing surface, and the ball’s movement is frictionless, among other conditions. In this case, the ball will not really ‘end’ up anywhere; it will continue indefinitely. However, does it ever return to its starting point? Does it eventually reach every area of the table? When we analyze this problem, we need to consider various factors such as the shape and size of the table, the angles at which the ball is struck, and the initial velocity of the ball.
When the direction or starting point of the ball is altered slightly, does the resulting path resemble the original one?
These inquiries are quite captivating from a mathematical perspective. The answers to these questions are not always readily available, particularly when the shape of the billiard is complex, such as a non-square or non-rectangular shape. For instance, in triangular billiards with angles of less than 100 degrees, it is established that periodic paths always exist—paths that the ball can follow and eventually return on themselves. This fact can be proven using mathematical reasoning. However, if one of the corners is modified to a slightly larger angle, no mathematician currently has the knowledge of whether the same is true or not.HTML format makes it difficult to convert the text as far as you have requested. Can you kindly provide the text in a different format?The real world continues to expand even further, according to Mazi Jalaal, co-author of the document and leader of the research group. Jalaal explains that many living organisms in nature have an external form of memory, leaving traces to remember where they have been. This information can be used to follow the same route again or, for example, to avoid exploring the same region when searching for food.
This led the researchers to an interesting idea: What if a new rule was added to the billiards game, stating that the ball may never cross its own previous path? The result, as shown in the figure below, is that the
The billiard table’s effective size diminishes as the ball becomes ensnared by its own path.
New and captivating inquiries
The trapping phenomenon adds to the intrigue of the system, turning even simple questions into compelling puzzles. How far does a ball travel before it becomes trapped? The answer varies based on the table’s shape, as well as the ball’s starting point and direction. At times, the ball only travels a few times the table’s size before becoming trapped, while in other instances it may travel 100 times that length. Where t …The question of how the ball ends up in its trapped state is quite complex. When the experiment is repeated on a computer numerous times, each time with a slightly different starting position and velocity, it generates stunning patterns of final configurations. The image above displays a few of these beautiful examples. Surprisingly, the resulting dynamical systems can exhibit chaotic behavior. A small change in the starting position or velocity of the self-avoiding ball can cause it to be trapped at a completely different point on the billiard. Unlike a typical billiard table, the self-avoiding ball behaves differently.Not every region has an equal chance of being the final destination. Some areas are more probable than others. Mathematicians have their work cut out for them in terms of explaining and proving these features.
Wide range of uses
An interesting aspect of the publication is that both of its first authors are master students. According to Mazi Jalaal, “The concept of a ‘billiard with memory’ is simple and novel enough that studying it doesn’t require years of experience. Thijs and Stijn did an excellent job of mastering the material and finding creative ways to address all of these new open problems. I’m thrilled that they are already able to…
Two researchers who were central to the completion of a piece of work.
The findings are just the initial stages of what could potentially be a completely new field of study. There are numerous intriguing mathematical inquiries that are now awaiting responses, as well as endless applications in physics, particularly biophysics. According to Jalaal, the concept of trapping is one that is ripe for exploration, even in real-world systems. For example, it is known that single-celled slime molds utilize self-avoiding paths. Do they also become trapped, and if so, what is the outcome? Or do they possess clever strategies to prevent this from occurring altogether? Perhaps they utilize trapping to improve their search techniques for food? The possibilities are endless.
The findings could enhance our comprehension of these biological systems, and maybe even apply the knowledge to improve the use of memory in robots for playing billiards.”
Journal Reference:
- Thijs Albers, Stijn Delnoij, Nico Schramma, Maziyar Jalaal. Billiards with Spatial Memory. Physical Review Letters, 2024; 132 (15) DOI: 10.1103/PhysRevLett.132.157101