How Many Times Can You Fold a Piece of Paper?
When I was a kid, I learned that there was a limit to the number of times a piece of paper could be folded. It was a lesson in exponential growth, the idea being that each fold doubles the paper's thickness, and even with something as thin as paper, quickly you'll end up with an unmanageable mess, too thick to fold further.
But the big question was always: Okay, so how many times can a given piece of paper be folded? In a brief third-grade science lesson we tried this experiment with various kid-sized pieces of paper, and often got to around six folds—and I just did it now with a large sticky note, and again got to six folds easily. Somebody (I can't recall whether it was our teacher or a fellow student) imparted the sage wisdom: seven folds is the most. This seemed plausible, because it seemed to hold up to all the testing a room full of savvy eight-year-olds could manage. Case closed: The universe only allowed for seven paper-folds on a given sheet. Oh, our minds would be blown in a few decades.
In January 2002, Britney Gallivan, then a junior in high school, folded a 4,000-foot-long roll of toilet paper to prove that 12 folds were possible (note that she used single-direction folding, given the long, narrow nature of her paper; my class had been using multi-directional folding, but still—wow). What's more, she did this after deriving a paper folding theorem (yes, it involves pi) that allows calculation of maximum folds based on paper thickness, length, and/or direction of folding, and accounts for the loss of usable paper at the edges due to the rounding that comes with extreme folding. That is some math magic right there, with empirical proof to boot.
Since Gallivan's proof, people have gotten up to quite a bit of fun with this. In 2007, the MythBusters tried the experiment and got nearly as far—but needed heavy machinery and used multi-directional folding, requiring a truly gigantic piece of paper to start with. Take a look:
Then in 2012, students at St. Mark's School in Southborough, Massachusetts visited MIT to attempt 13 single-direction folds. They didn't actually use Gallivan's single-sheet method, instead choosing to layer the first 64 sheets (equivalent to six folds) on top of each other and then begin the folding, but this is still a lot of fun:
For more on Gallivan's achievement (and the math), read this page from The Historical Society of Pomona Valley.
See also: Folding Space-Time Using a Music Box