January/February 2022 27ILLUSTRATION BY ALYSE MARKEL
it involves rules of motion that do not change, lead-
ing 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 Br yn Mawr College, researches the
chaotic properties of dynamical systems, includ-
ing 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 paral-
lelogram-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 tar-
get ball and the pocket, that is the easiest shot you
can make, Donnay says, because it’s the least cha-
otic 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 condi-
tions—you might call it the butterf ly 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 thesystem. 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 ricochet-
ing after contact with the target ball, but beware:
“The more complicated the options, the more dif-
ficult 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 mathemati-
cally 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. Ever yone has innate
mathematical abilities.”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 http://www.youtube.com/watch?v=LsvJSRrwpio.Is It Possible to Build a Can’t-Miss
Pool Table?