Folding a Paper in Half... Repeatedly

I remember the dare from childhood that said nobody can fold a piece of paper 7 times. I tried and failed, of course. Since nobody else seemed able to do it either, I assumed it had to do with strength (or rather, the lack of it).

 

But that’s not the answer, explains Steven Strogatz in The Joy of x. The reason is more prosaic:

“Each folding approximately doubles the thickness of the wad, causing it to grow exponentially (1, 2, 4, 8, 16…). Meanwhile, the wad’s length shrinks in half every time, and thus decreases exponentially fast (1, ½, ¼, 1/8…)”

That much is obvious. But here’s the next step in the train of thought that most of us don’t think of:

“For a standard sheet of notebook paper, after seven folds, the wad becomes thicker than it is long, so it can’t be folded again. It doesn’t matter how strong the person doing the folding is… (It) can’t happen if the wad is thicker than it is long.”

 

Britney Gallivan, in high school, derived the formula for how many times a piece of paper can be folded!

Where L = length of the paper, T = thickness of the paper, n = number of times it can be folded.

 

She then went and bought a really long of toilet paper, unrolled it, and with the help of her parents, “smashed the world record by folding the paper in half twelve times”!

 

Who’d have thought this was a maths problem, not a strength problem?!