26 August 2008

How many times?

How many times can you repeatedly fold a sheet of paper in half? It's widely accepted that about six or seven times is the maximum possible, and a quick experiment with a piece of writing paper, a sheet of newspaper, or any normal paper you can find around the home will prove that this is correct. Or will it? What does 'correct' mean? What does 'proof' mean?

A mathematician will tell you that however many times you do the experiment and find you can't fold the paper a seventh time, that is not proof. You cannot prove something to be impossible, only that something is possible. Folding a piece of paper six times and failing to fold it seven proves that six is possible, but not that seven is impossible.

Remarkably, someone has managed to fold a piece of paper twelve times! Was there something special about this piece of paper? Yes and no.

The paper was a long roll of toilet paper. The relevant attribute of this paper was not that it was especially thin (try folding a single sheet of toilet paper yourself) but that it was especially long.

Britney Gallivan, a high-school student from California, was not prepared to take 'no' for an answer. She began by developing some mathematics for paper folding, and this showed her that a piece of paper that is long enough can be folded many more times along its width than a shorter piece. Armed with this knowledge she did the experiment - and managed to fold it twelve times.

There are several lessons to be learned from this.

What seems to be impossible may, in fact, be perfectly possible if we go about it in the right way. Technology has shown this to be true over and over again. Here are a few things that were once thought to be completely impossible - travelling to the moon, ships made of iron, building a flying machine, sailing round the world, the earth moving, continents moving, orbiting a satellite.

Common sense often lets us down. It would be a wonderful thing to learn the value of not making assumptions or jumping to conclusions. But we are designed to assume and conclude, this serves us well most of the time and enables us to deal relatively simply with a very complex world.

Britney Gallivan's paper folding achievements are described online. I encourage everyone to read them, if mathematics is not your forte you can skip that part, but please understand that it was the mathematics that led her to a simple, elegant, but entirely unexpected conclusion. With hindsight it seems obvious, but nobody had thought of it before Britney. Clever young lady!

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