So help me out here. I was leafing through Malcolm Gladwell’s “The Tipping Point” recently, and at the very beginning, in the introductory chapter, he writes this, as a way to illustrate the “geometric progression”:
“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.”
Hmm. That seemed like a very odd example, when I first read it. I get the point he’s trying to make, about the often astonishing nature of geometric progressions — indeed, half of the book I just wrote is about Jefferson’s (many) insights into the nature of geometric scaling and geometric growth. I get it: each time you fold the paper, the stack doubles in size, and that after 50 folds it will be 2^50 — two raised to the fiftieth power — sheets tall, and 2^50 is a really really big number. If you start with a piece of paper .01 inches thick, after 50 folds it will be (.01)*(2^50) inches tall — 11,200,000,000,000 inches, or around 177,000,000 miles.
But still, something seems off to me. 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.
That’s awfully small . . . Maybe some of you out there know more about all this than I do and can help out — is a single molecule of paper (and is there even such a thing as a “single molecule of paper”??) smaller than that? And am I wrong that, with all the terrific examples out there of the power of geometric increase (read my book!), this is a really lousy one?