Pool is as much a battle against chaos as it is a battle between two opponents. Unless you’re a pro, it’s likely you’ll misalign your cue stick with the cue ball, or miscalculate the angle for your next shot. As your errors compound, the conditions on the table become less predictable.
But a secret mathematical weapon known as a dynamical system could help you dominate the table. These systems function under a specific, consistent set of laws over time, meaning you can master them through study, observation, and experience. Researchers use them to build predictive models for things like planetary orbits and the path of a viral disease through a population. There are two primary types of dynamical systems: deterministic and stochastic. Pool is a deterministic dynamical system because it involves rules of motion that do not change, leading to consistent consequences for each way you shoot the cue ball. Stochastic dynamical systems involve changing rules, making them more random or probabalistic.
Victor Donnay, a mathematics professor at Pennsylvania’s Bryn Mawr College, researches the chaotic properties of dynamical systems, including those illustrated by billiards. He says you can make the chaos of a pool table more predictable if you learn the table’s deterministic rules of motion.
▶ Rectangular tables are the least chaotic.
The shape of the table determines whether you’re playing with a simple dynamical system or a complex one. Rectangular tables are the most common for a reason, Donnay says. “It’s simple enough that we can understand what’s going to happen and predict the future.”
A rectangular table’s sides will bounce shots away at an angle equal to the angle at which the shot arrived. This leads the cue ball to traverse a parallelogram-like shape if uninterrupted by another ball—if it bounces off two sides, its trajectory will be parallel to its trajectory just after your strike.
▶ Straight shots mean less chaos.
If the cue ball lands in a straight line with your target ball and the pocket, that is the easiest shot you can make, Donnay says, because it’s the least chaotic option on the table. More bounces in your shot equates to a less predictable outcome.
▶ Fewer bounces mean less chaos.
If there are no straight shots left, seek out balls you could sink with one bounce against the side. Each bounce magnifies the small angular errors you might have made when hitting the cue ball. Donnay calls this sensitive dependence on initial conditions—you might call it the butterfly effect. “It’s one of the main mathematical meanings of chaos,” Donnay says. “A small change—how you hit the ball—could lead to a big change later on.”
Shoot directly at target balls instead of trying to hit them off a bounce from a side, if possible. The latter approach adds more potential for error. “Unless you’re super precise in how you hit the [cue] ball at the beginning, your likelihood of hitting the target ball in the way you intended would get smaller as the error grows,” Donnay says.
▶ Create chaos for your opponent.
Don’t have a feasible shot? Play defense, and force your opponent to deal with more variables in the system. Tapping the cue ball so it’s tucked against a side, behind a ball your opponent can’t hit, or far away from the target creates greater potential for compounding errors by adding bounces, sharper angles, and distance. Donnay says this line of thinking—forgoing your own shot to play defense—is a type of mathematical risk analysis.
▶ Embrace the chaos, spin the cue ball.
So far, we’ve assumed that you will hit the ball head-on with the cue stick. But if you apply spin to the cue ball by hitting it off-center, the new motions will create a more complicated dynamical system. Backspin, achieved by hitting the cue ball just below center, can keep the cue ball from ricocheting after contact with the target ball, but beware: “The more complicated the options, the more difficult it is to control them,” Donnay says. Practice until you can do it consistently.
Donnay admits that it’ll take time to master the pool table’s rules, but thinking mathematically can help you improve. Pro pool players likely don’t think of themselves as mathematicians, he says, but they use their visualization skills in place of formal formulas. “As creatures, we’re designed to look for patterns in the world. Everyone has innate mathematical abilities.”
Is It Possible to Build a Can’t-Miss Pool Table?
An elliptical pool table design from YouTuber “The Q” claims to make you sink your ball every time. But Donnay says this assertion is partly false. “Saying that you’ll always go in the hole is a little misleading if you don’t specify where you begin [on the table],” he says.
But The Q found the one foolproof place on the table to begin your shot. An ellipse has two points on its major axis, called foci; if you trace a ray from one focus until it bounces against the edge of the ellipse, it will always pass through the other focus as a result of the bounce. The Q’s pool table has the pocket positioned at one focus, so if you set up your ball at the other focus and hit it, per our diagram, the ball will land in the pocket no matter the angle of your strike. To see The Q’s table, head to www.youtube.com/watch?v=LsvJSRrwpio.