The folding limit: paper runs out of length, not strength

You have heard that paper cannot be folded more than seven times. Try it. Around the sixth or seventh fold the sheet simply stops. Everyone says the paper got too strong. That is not what happened. The paper ran out of length. Every fold does two things at once. It doubles the thickness, and it halves the length you have left. Six folds into an A four sheet, the stack is six point four millimetres thick and under five millimetres long. It is now taller than it is wide. And here is the part nobody draws. A fold is not a crease. It is a bend with a radius, and the outer layer has to go the long way around it. Bend a stack of sixty four layers and the outside runs twenty millimetres longer than the inside. You have four point six millimetres of paper left. There is nothing to spend. In two thousand and two a high school student named Britney Gallivan worked out exactly what that bend costs, fold after fold, and summed it. One line. For a sheet of thickness t, folded in half n times in the same direction, this is the length you need. Run it. Four folds wants just under sixteen millimetres of paper. Six folds wants two hundred and twenty four. Seven wants eight hundred and seventy eight — most of a metre, for one more fold. Every fold costs about four times the one before, because the thickness that has to bend just doubled. So the honest answer for the sheet on your desk. Two hundred and ninety seven millimetres of A four buys you six folds. Six. Not seven. And seven was never a law anyway. The formula says twelve folds needs eight hundred and seventy nine metres of paper, so she went and got eight hundred and seventy nine metres. A one point two kilometre roll, down a shopping mall corridor, folded twelve times. The limit is real. The number is not. It was never about how strong the paper is. It is about how much of it you brought.