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Folding a piece of paper in half

By David Shaw, 29 October 2013

This activity sounds super easy from the title, but there’s more to it than you might expect!

You will need

• Sheet of A4 paper
• Roll of cheap toilet paper – the thinner the better

What to do

1. Before you start, try answering this question: How many times do you think you could fold a piece of paper in half?
2. Take the sheet of A4 paper and fold it in half.
3. Then fold it in half again, and again, until you can’t fold it in half anymore.
4. Remember how many times you folded it in half. Was it more or less than you expected?
5. Find a long space to roll out the toilet paper – if it is very calm, you might be able to do this outside.
   

   Find a long place to unroll your toilet paper.

6. Roll out as much paper as you can.
7. Take one end of the paper and bring it back to the other end.
8. Straighten the folded piece of paper.
9. Now, take the folded end of the paper and bring it to the open ends of the length of paper. Straighten out the paper again.
10. Keep folding the paper in half in this way, until you can’t fold it in half any more.
11. How many times did you fold it in half? Was it more or less than you expected?

What’s happening?

There are two things that are happening when you fold paper in half, and they both mean that the more folds you’ve made, the harder it is to fold again.

Every time you fold a piece of paper in half, its area is halved. If you start out with an A4 sheet, then after four or five folds, it starts getting quite small. If you could fold it in half nine times, it would only be about one square centimetre in area!

The other problem you run into is that the paper keeps getting thicker. A piece of paper is very thin – the standard sheet of A4 paper is about 1/10 of a millimetre thick. But every time you fold the paper in half, you double the thickness. So after two folds, it’s four times as thick as a single sheet, and after three folds, it’s eight times thicker. After nine folds (if you make it that far) it will be 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 512 times thicker. If a single sheet of paper is 1/10 of a millimetre thick, then after you folded it nine times, it would be over five centimetres thick!

Applications

It is often said that it’s not possible to fold a piece of paper in half more than seven times. In 2002, a high school student named Britney Gallivan proved it was possible to do a lot more than seven folds. Britney was challenged by her teacher to fold a sheet of paper in half 12 times. To help her understand the problem, she came up with two formulas working out how much paper you need to start with. One worked by folding in different directions, like you would with a sheet of paper, and one where you keep folding in the same direction, like you would with a roll of toilet paper.

Her formula said that she needed a sheet that was a lot wider than it was thick, so she tried using a very thin sheet. Paper doesn’t get much thinner than 1/10 of a millimetre, but gold comes in sheets that are only a few atoms thick. Starting with a square about ten centimetres on each side, she managed to fold the gold 12 times.

When she presented her solution to her teacher, the teacher said that she had to fold a piece of paper, not a sheet of gold. In order to make 12 folds in a piece of paper, she had to find a special roll of toilet paper that cost $85 and was 1.2 kilometres long. After seven hours folding, she eventually managed to get her paper folded in half 12 times.

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