Thursday, April 14, 2016

Paper Folding to the Moon

If you folded this piece of paper in half, it would now be twice as thick as it was before:

So my question is this: how many times would you have to fold this paper onto itself to reach the Moon? I’ll give you a chance to guess, so pick the closest one from the options below.

Well, let’s see how we’d figure it out. I don’t know how thick one piece of paper is, but I know it’s pretty thin. I can, however, estimate how big those 500 page reams are. They’re about 2 inches high, so maybe that’s about 5 cm. That means one page is about 0.01 cm high. And what of the Moon?

Mean distance from the Earth is about 384,000 km, or about 3.84 x 1012 pages away. So you’d expect that you’ll need an awful lot of foldings to get there, right? Well, hang on for a second.

When I start with an unfolded page (zero foldings), it’s one page thick. When I fold a page once, it will be 2 pages thick. But — and this is key — when I fold it twice on itself, it’s not three, but 4 pages thick.

If I fold it a third time, I’ll see that it’s 8 pages thick. Can you see a pattern here? Paper folding is exponential, so that if I fold it a fourth time, it’ll be 16 pages thick (so that option is clearly wrong), a fifth time will give me 32 pages thick, and so on. By time I get to 9 foldings, my folded paper is bigger than my original ream of 500 sheets. By time I get to 20 foldings, my folded paper is more than 10 kilometers high, which surpasses Mt. Everest. 41 foldings will get me slightly more than halfway to the Moon, so that means that 42 foldings is all it takes! (Of course, good luck folding a real piece of paper more than 7 or 8 times…)

Pretty incredible, isn’t it? But that’s the power of an exponential, that it lets you turn small things into huge things by simply compounding what you have over and over again. And incredibly, it only takes 42 foldings of a paper to get from the Earth to the Moon, and only about 94 foldings of a paper to make something the size of the entire visible Universe! And how surprised are you that the answer is so small a number?

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