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Essentially, just as a circle has an exact center point, an ellipse has two center or focal points. Like many claims on the internet, Donnay says this assertion is partly true and partly false. This explains why pool balls knocking into one another behave more chaotically than when they bounce off the wall.
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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.Īnd such systems can only get more chaotic as you add more elements or obstacles to it, Donnay says. “Rather than having a nice, repeatable pattern where we can kind of see what’s going to happen… will start bouncing all over the place,” Donnay explains. It’s also known by a few other fun names, like “discorectangle,” “obround,” and “sausage body.”
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If you’ve ever been to a professional football game or skated at a hockey rink, you’ve encountered the stadium shape. “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.īut 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.
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Table Stakes: How To Build a Simple Poker Table 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. Mathematicians like Donnay seek to understand the geometrical properties of the dynamical system’s trajectories and long-term behavior.
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Often it involves differential equations, according to the math whizzes at the University of Arizona. In mathematics, dynamical systems is the branch of study focused on systems that are controlled by a specific, consistent set of laws over time. “You can have deterministic ones, or slightly stochastic or random ones, but billiards is the deterministic one where there’s a rule of motion.” “Billiards fits into a more general classification in mathematics of dynamical systems,” explains Donnay. “Excellent billiards players don’t think of themselves as mathematicians.”īut 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.