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An intellectually challenging game of loop

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Media captionAlex Bellos had a pool table constructed in the shape of an ellipse to prove a point about geometry

Maths writer Alex Bellos recently spent a lot of money on a pool table - an unusual one with only one hole, and no straight sides. It is, in fact, in the shape of an ellipse. Why would anyone do such a thing?

Geometry and maths are at the heart of any game of pool. On a standard rectangular pool table, a ball will bounce off the cushion at the same angle it arrived. Add a bit of spin to the ball and you can alter the ball's path, but this basic principle is fundamental to any pool player's game.

This was already understood in ancient Greece. The mathematician Euclid realised that light travelled in a straight line and reflected off a shiny surface at the same angle it had struck it.

It is another Greek mathematician, Apollonius, who is credited with discovering that an ellipse is the shape you get if you cut through a cone at an angle - a shape like a circle but narrower around the middle and longer at the ends.

Image copyright Science Photo Library
Image caption The red shape - the intersection of a cone with an inclined plane - is an ellipse

All ellipses have an interesting property - they have two focus points.

Imagine standing in an elliptical room with one long curved mirror along the walls. If you were standing on one focal point and you shone a torch at the wall, it would reflect off the wall and cross the other focal point.

Something similar happens in "whispering galleries". If you stand at one of the focus points in a room with elliptical walls or an elliptical ceiling and talk quietly, the sound waves will bounce off the roof or walls in such a way that a person at the other focus point will be able to hear what you are saying. There are examples at Grand Central Station in New York and St Paul's Cathedral in London.

Image copyright AFP
Image caption The Whispering Gallery in the dome of St Paul's Cathedral

And this explains why Alex Bellos asked someone to make him an elliptical pool table.

"The reason I built it is maths textbooks always say - imagine you built a pool table in the shape of an ellipse; whenever you put it on one focus point it will rebound to the other focus point," says Alex.

But the theory, it seems, is better than the practice.

When Alex first got the table he put his ball on the spot, closed his eyes and hit it against the cushion. The maths says it should just roll into the pocket.

It didn't. Disaster!

"My heart sank," remembers Bellos. "I thought, 'My God, the guys who built it have built it wrong!'"

But he measured the table and it was a perfect ellipse. Hitting a pool ball on an elliptical pool table is just much more difficult in real life - you have air resistance, friction, the irregular surface of the baize, and other factors such as how hard you hit the ball and where the cue makes contact with it.

"This is almost an example, not of mathematics but how mathematics changes when it becomes physics," says Bellos.

Hitting one ball around the table and watching the geometry in action is mildly entertaining - when you hit it cleanly at the right speed it will often fall into the hole.

But Bellos wanted to make it more interesting. What he did was to take the geometry and invent a game - not pool but Loop.

Unlike pool you only have four balls - the cue ball, one black and two colours, one for each player. The idea is the same - if you pot your colour, and then the black, you've won.

But Alex has a tip: "Forget about the sides - it's all about the focus points."

To pot the balls you have to use the geometry. The two focus points of the ellipse are still crucial to potting the balls and the principle of hitting the ball from one focus point (marked by a spot) to the other focus point (where the hole is) is still central to the game.

The ideal way is to play is to use the white ball to direct the coloured ball over the spot on the first focus point. Then, when it rebounds off the cushion, it should - other things being equal - drop into the pocket. It's not easy to explain on the page but if you watch the video above you should get the idea.

It's surprisingly fun to play but it's still difficult to know how much practice you would need to make it a game you could win every time - and whether a professional with an instinctive grasp of the geometry of a rectangular table would get it instantaneously… or find it completely baffling.

Image copyright Getty Images

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