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Don’t you just love proving people wrong by doing what is supposed to be impossible? My own personal triumph was beating the Saltine myth – I was once told that it was impossible to eat seven Saltines in a minute, and after a few attempts, I was able to prove them wrong (as well as ensure that I’ll never want to eat a Saltine again). I then proceeded to lick my elbows and say “toy boat” ten times quickly. It’s all possible, even though many don’t believe it.

However, there’s one challenge I have never been able to figure out – how to fold a piece of paper more than eight times. I have heard it stated as fact that one cannot fold a paper in half more than eight times, because the doubled and re-doubled paper quickly becomes too thick. This reasoning had always seemed pretty odd to me – what was this magical property of hardened tree pulp that caused it to stop folding after so long? Thanks to a student named Britney Gallivan, it turns out that this impossibility is just as mythical as not being able to lick your elbows. By developing her own formulae for paper folding, she calculated how much paper one needs to achieve any number of folds (and herself managed to break world records by folding a piece of paper twelve times).

It all started Gallivan’s junior year of high school in 2001 when she was given an extra credit challenge for math – fold a piece of paper twelve times. (Don’t you love it when teachers give you extra credit for what they assume is an impossible task?) She figured that either the paper had to be quite wide, or very thin; she went with the latter for her first attempt. Using some precise tools, she was able accomplish the feat using gold foil, which is only eleven millionths of an inch thick. Of course, this was not good enough for the teacher, who insisted that something with of the thickness of paper should be used for the challenge.

Back to the drawing board, Gallivan now had to calculate how big of a sheet of paper she needed in order to fold it twelve times. First, she specified what a paper with n folds was, based upon the layers seen in the paper. One fold would have two layers, two folds would have four layers, three folds would have eight, and so on; in other words, n folds meant 2^n layers.

She also determined the equations for how much paper was needed for folding, based upon the way paper folds. She came up with equations for both alternating and single-direction folding (alternating is what some people call “hamburger-hotdog” folding, whereas single-direction keeps folding the same point over and over again).

For single-direction folding, the equation is:

Single directional folding

Where L is the minimum possible length of the material, t is material thickness, and n is the number of folds possible in one direction. L and t must be the same unit.

For alternate-directional folding, the equation is:

Alternating direction folding

Where W is the width of a square piece of paper and n is the number of paper folds. This equation is not exact, but gives a close approximation of the limits for paper size.

If one looks at her equations and the thickness of a basic 8.5″ by 11″ paper, it makes sense that it was thought that folding a paper more than eight times would be impossible. However, instead of being impossible, it’s merely a problem that gets exponentially more difficult with each iteration. Think about the story where a king pays an artist a penny for his first day of work, two pennies on the second, and so on – by the end of a month, the artist is making more than a million dollars a day. The paper folding problem is equally distressing as the king’s future debt to the artist.

Ultimately, Gallivan discovered that for a large number of folds, single-direction folding required less paper. However, in order to achieve her teacher’s required twelve folds, Gallivan would need a 4,000 foot-long piece of paper. She eventually found a roll of special toilet paper that matched the job, and set out to break a record.

Unrolling the paper in a mall at her hometown in Pomona, California, she and her parents went to task. After seven long hours of folding their hearts out, the three successfully executed twelve folds, shattering the previous world record. For her efforts, Gallivan received her extra credit, and in the process she changed the world in an indiscernible but distinct way. As for the generations of people who have asserted that such a feat is impossible, Britney proved them wrong once and for all. It’s amazing what you can accomplish with a little imagination, a double-helping of ambition, and an industrial-sized roll of toilet paper.

Further reading:
Historical Society of Pomona Valley article on Britney Gallivan
Wolfram Mathworld on the Mathematics of Folding

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