Geometry

# Corner Pocket? Not When You Play Pool On An Ellipse

To win a game of pool, you have to know a bit of geometry — a surprising fact for a sport usually played after hours in a dim, smoke-filled bar. That wasn't enough math for Alex Bellos, however. Figuring that rectangles are for squares, he invented a pool table shaped like an ellipse. Corner pocket? Try left focus.

## The Alluring Ellipse

You can think of an ellipse as a squished circle. While a circle has one focus point, right in the center, an ellipse has two. There's a particular formula you can use to find the foci of an ellipse, but for our purposes you can visualize them this way: imagine you have two pins sticking out from a wall, and a long loop of string hanging from the both of them. If you use a pencil to pull that loop taut, then use the string as a guide to draw a shape around the pins, you'll end up with an ellipse that has those pins as its focus points.

Because that string never changes length, it shows that the sum of the distances between the two foci and any point on the ellipse is always the same. If the ellipse was a wall of mirrors, you could shine a light from one focus at any point on the wall, and it would always cross the other focus.

## You Can't Spell Loop Without Pool

Strange properties like these are what led Bellos, a British journalist who covers sports and mathematics, to create his unusual pool table. Math books are always explaining ellipses by describing how balls would behave on an elliptical pool table, so he decided to create a real one — and a game to go along with it, called Loop.

To play, you need a cue ball, an 8-ball, and one colored ball per player. Like in pool, you use the cue ball to hit the colored balls into a pocket. But unlike pool, there's only one pocket, and it isn't on the edge; it's at one focus. To sink the ball, you need to send it at an angle that would pass through the other focus, which is marked as a dot on the table.

In theory, any ball that takes that angle will always land in the pocket. In practice, it's a lot harder. The pocket isn't very big, and variations in speed can make the ball jump the pocket or stop short. "This is almost an example, not of mathematics but how mathematics changes when it becomes physics," Bellos told the BBC. Challenging as the game is, there's already been a world championship — and a world champion, Cambridge professor David Spiegelhalter. If you want to take him down, you'll have to practice. Luckily, Bellos sells made-to-order Loop tables on his website.

– Numberphile
Curiosity Staff