Over at the legal blog Volokh Conspiracy, David Post asks about this quote from Malcolm Gladwell’s book “The Tipping Point”:

“Consider, for example, the following puzzle. I give you a large piece of paper, and I ask you to fold it over once, and then take that folded paper and fold it over again, and then again, and again, until you have refolded the original paper 50 times. How tall do you think the final stack is going to be? In answer to that question, most people will fold the sheet in their mind’s eye, and guess that the pile would be as thick as a phone book or, if they’re really courageous, they’ll say that it would be as tall as a refrigerator. But the real answer is that the height of the stack would approximate the distance to the sun. This is an example of what in mathematics is called a geometric progression.“

Post points out that this is a very odd example. Geometric progressions can be astonishing but this stretches credibility. It would seem to require that folding somehow increase the amount of matter in the paper.

Folding can’t increase the amount of matter in the paper, and it just seems wrong to think you can somehow, magically, stretch a single piece of paper to the sun by folding it enough times. With each fold, the stack gets taller, but it also gets smaller (in area) — and also by a factor of 2 with each fold. So the initial area is reduced by a factor of 2^50. At some point — the size of an individual molecule?? — you can’t (even in theory) fold it any more. If your initial piece of paper is, say, 24 inches square (576 square inches), you would end up with a stack with an area of 576/(2^50) inches square, .000000000000511 square inches, or 5.1 * 10^-13.

Post’s gut feeling here is right. No piece of paper can be folded 50 times. And the limit is not, in fact, anywhere near the size of a single molecule or atom.

For a long time, there was a fairly common assumption that a piece of paper could not be folded in half more than 8 times. In fact, it is quite hard to fold any common piece of paper in half more than seven times. Go ahead and try it.

But this all changed thanks to a high school girl from California named Britney Gallivan. Here’s how Gallivan’s Facebook page describes her accomplishment:

In January 2002, while a junior in high school, Gallivan demonstrated that a single piece of toilet paper, 4000 ft (1200 m) in length, can be folded in half twelve times. This was contrary to the popular conception that the maximum number of times any piece of paper could be folded in half was seven. Gallivan succeeded in folding a very long sheet of toilet paper in half 12 times. She calculated that instead of folding in half every other direction, the least volume of paper to get 12 folds would be to fold in the same direction, using a very long sheet of paper. A special kind of $85-per-roll toilet paper met her length requirement. Not only did she provide the empirical proof, but she also derived an equation that yielded the width of paper or length of paper necessary to fold a piece of paper of thicknesstanynnumber of times.

Here’s a link to a discussion of the formula for determining the limits of folding.

Geometric progression may be astonishing. But nature imposes limits on even the most aggressive origamist.

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