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Maths of Planet Earth | Limitless Applications

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Published on June 3rd, 2014 | by Liam

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Can seven cylinders touch together?

Many years ago (before we knew cigarettes were so bad), a maths reporter posed a simple problem. Maths+StatsMartin Gardner had been playing with cigarettes on his desk, seeing how they touched each other. He wanted to know if seven cylinders could be arranged so each cylinder touched all the others.

Martin eventually came up with a solution for cylinders of a certain length. But he wasn’t completely satisfied – what if the cylinders had no ends? What if they were infinitely long?

Many years later, a team of mathematicians examined Martin’s problem. To begin with, they needed a mathematical way of describing a cylinder. First, they decided their cylinders would be 2 centimetres (cm) wide. Then they imagined a straight line running down the centre of the cylinder. Every point in the cylinder was within 1 cm of the centre line.

On a flat piece of paper, two lines drawn in different directions will eventually meet somewhere, as long as the paper and line are long enough. But in 3D space, they can get close together without intersecting. This was important – the researchers wanted their cylinders to touch each other, without colliding.

To have the cylinders touching, the two centre lines needed to be exactly 2 cm apart. If they stayed too far apart, say 3 cm, then the cylinders never touched. If the lines got too close, say 1 cm, then the cylinders collided and part of one cylinder was inside the other. Only at 2 cm would there be exactly one point where the two cylinders touched.

The mathematicians used this understanding to write Martin’s original question as a set of equations. Then they put the equations into a computer and waited for several weeks. When the calculations finished, they found two solutions!

This isn’t the end for this cylindrical conundrum. Can mathematicians find a solution for eight cylinders? And if not, can we prove that it’s impossible?

More information
A tale of touching tubes
Coverage from the Gathering for Gardner conference

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