The anticipation in the room is palpable as you methodically chalk the end of your pool cue and lean against the side of the table, lining up your shot against the lone, surviving 8-ball. You test the agility of your cue, back and forth, considering the best plan of attack to sink the ball and take home the glory.
Summoning the courage of a base jumper, you strike the cue ball and watch as it ricochets off the table’s edge, slamming the 8-ball clean into the opposite pocket.
Unfortunately, many casual games of pool—and even professional ones—don’t end this neatly. Billiards, which traces its origins from a 15th-century English lawn game, is as much a battle between two opponents as it is a battle against chance, whether you’re playing 8-ball pool, English snooker, or any other of the countless pool iterations.
But what if there were a mathematical secret weapon to help you play better and impress your friends? According to Victor Donnay, a mathematics professor at Pennsylvania’s Bryn Mawr College, there just might be, and it involves something called a “dynamical system.” Donnay’s research focuses on the chaotic properties of dynamical systems, including those modeled on billiards.
“Billiards fits into a more general classification in mathematics of dynamical systems,” explains Donnay. “You can have deterministic ones, or slightly stochastic or random ones, but billiards is the deterministic one where there’s a rule of motion.”
In mathematics, dynamical systems is the branch of study focused on systems that are controlled by a specific, consistent set of laws over time. Often it involves differential equations, according to the math whizzes at the University of Arizona. Mathematicians like Donnay seek to understand the geometrical properties of the dynamical system’s trajectories and long-term behavior. The applications extend beyond pool, of course: The study of dynamical systems allows researchers to model a wide variety of phenomena, such as the motion of planets in our solar system, the way plants grow, and even how diseases spread through a population.
With that in mind, Donnay says that no matter how much of a pool shark you might be, there’s a fundamental mathematical basis that sets an easy billiard table apart from an impossible one: namely, shape.
“Square or rectangular is one of the simplest shapes. That’s the one people use for real billiards,” says Donnay. “There’s a lot of regularity of what’s going to happen…so that’s a very simple and easy-to-understand dynamical system.”
Here’s how that rectangular pool shot plays out: It will bounce off the table’s edge with an angle equal to the angle it arrived at and traverse a roughly square shape, keeping parallel lines all the way. A circular table is a little more complex, says Donnay, traversing a starlike pattern instead of a square one. Nevertheless, the billiard ball’s trajectory is predictable from a mathematical perspective.
“In both of these cases, it’s simple enough that we can kind of understand what’s going to happen and predict—with a little drumroll—the future,” he says.
But things get tricky when you flatten out the edges of that circle, Donnay says, into a shape that mathematicians call a “stadium.” In geometry, a stadium consists of a rectangle with two semicircles on opposite ends. If you’ve ever been to a professional football game or skated at a hockey rink, you’ve encountered the stadium shape. It’s also known by a few other fun names, like “discorectangle,” “obround,” and “sausage body.”
“Rather than having a nice, repeatable pattern where we can kind of see what’s going to happen…[the ball] will start bouncing all over the place,” Donnay explains. “This is an example of chaotic motion.”
Mathematicians have found that concave curves, like the edges of a stadium, as well as convex curves, like the circumference of a ball, can create this kind of chaos.
And such systems can only get more chaotic as you add more elements or obstacles to it, Donnay says. This explains why pool balls knocking into one another behave more chaotically than when they bounce off the wall.
Avoiding this kind of chaotic motion when possible, and keeping an eye on the angle you strike your cue ball with, can help improve your game. But it won’t necessarily make you a pro overnight, Donnay warns.
“It may be that excellent billiards players don’t think of themselves as mathematicians,” Donnay says, “but they have actually developed a mental way of thinking about it that we would think of as mathematical.” But instead of formulas, he says, they operate on the basis of practice and intuition and visualization.
If you’re really out to impress your pool hall friends, you could try building a loss-proof pool table, based on a recent design demonstrated in a video from YouTuber “The Q.” The video (below) shows how to build an elliptical pool table with rubber sides, which will supposedly help you sink your ball every time.
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Like many claims on the internet, Donnay says this assertion is partly true and partly false. Essentially, just as a circle has an exact center point, an ellipse has two center or focal points. If you shoot the ball—in any direction—from one of the focal points, it will end up in the other. For context, an ellipse is just a circle that has been stretched in one direction to give it an oval shape. (Not all ovals are ellipses, though).
“The cool idea in the video is they put the hole at one focal point and they start the shot at the other point,” Donnay explains. “There are lots of places you could start at the table, and directions you can go that are not going to go into that hole. So saying that you’ll always go in the hole is a little misleading if you don’t specify where you begin.”
While Donnay plays the occasional game of pool, he says he’s more interested in how dynamical systems can be used to model large-scale phenomena like climate change or epidemiology. If you can imagine a thriving climate as a predictable, circular table, Donnay says, then rising carbon dioxide levels shift the climate toward a less predictable (and chaotic) table shape.
Math can often be abstract and seem distant from our lived experience, but Donnay hopes that understanding the connection between your neighborhood pool table and larger concepts, like climate-change-driven flooding and heat waves, will help everyone better appreciate how serious our situation is.
“Sometimes a small change from the circle to the stadium can produce a big change,” says Donnay, referring to something mathematicians call a tipping point, or bifurcation. “Maybe we overfish just a little bit and the fish go extinct, [or] the temperature goes up a little bit more, and suddenly we get these new effects, like more melting of the glaciers that are going to cause a big change that we can’t control anymore. I’m hoping that this becomes something that everybody in society understands and learns about.”
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