I recently rediscovered James R. Newman‘s four-volume The World of Mathematics in a thrift store. For 25 cents a piece, I got volumes i=1,4 where i <> 3; that is: I got volumes 1, 2, and 4. This was a disappointment, given that Volume 3 included Nagel and Newman’s article on Goedel’s Incompleteness Proof. So I got that volume from the library, with the thought of revisiting this article. However, by the time the book arrived, I had thought quite a lot about the proof (well, there are actually two of them, but let’s not go there), and I came to the conclusion that I’m just not that interested in it any more. As important an intellectual accomplishment as it is, the fact remains that it doesn’t really matter very much, even in mathematics—and given my postmodernist frame of mind, it just seems kind of obvious.
In going through the volume, however, I came upon a nice (and easy) article by Edward Kasner (the guy who coined the term “googol”, which the guys at Google misspelled, but maybe for the best) and Newman: Paradox Lost and Paradox Regained. In it, they present the following apparent paradox involving two quarters:
The interesting thing about this paradox is that the authors apparently felt it so trivial as to not even explain it. They do go on to explain the mathematics behind it, but how it is applied is not explained and it is all rather more complex than it need be. This is not to say that I went right along with them. After finishing the article, I spent about an hour thinking about this problem without finding a solution. It was only after waking up at 3:00 am that the solution hit me. Sleep is usually helpful in such matters.
This is exactly the kind of math problem that most people hate:
- It takes a bit of thinking just to see the problem
- It is infuriating once you’ve seen the problem
- It is a totally irrelevant problem
And then there are people like me who are nerds. We are just like everyone else in thinking that such problems are hard to understand and answer and pointless to boot. But like Vladimir and Estragon, we know you have to fill your day with something, and this will do as well as anything else. (Better in fact; reality TV is really beginning to bug me; and I only see snippets of it, like the court shows. I suppose what really bugs me is the fact that other people find such debased entertainment, well, entertaining.)
And Now Back to Our Quarters
Why does this seem paradoxical? I will try to explain (actually, I now find the problem harder than the solution). If you were to roll a quarter along a straight line segment equal to the length of the quarter’s circumference, it would be oriented the same way at the end as it was at the beginning. If you did the same thing along a line segment of half this length, the quarter would be oriented upside-down. This should all be obvious. If not, try it out yourself. Or just think about it; don’t expect me to do everything…
The quarter rolling around the other quarter is equivalent to the second line: half the circumference. Thus, the quarter should finish upside-down. But it doesn’t; this is because the two cases are only equivalent in terms of the distance rolled; the displacements (directional distance) are quite different. This, in itself, does not answer the question, however.
When you start rolling the quarter, say from 0 to 10 degrees (obviously, this works with infinitesimal elements; I will leave the Calculus to you) you are also rolling it from 0 to 10 degrees on the stationary quarter. Thus, while the rotating quarter does indeed revolve only 180 degrees from start to finish, the axis on which it is rotating is similarly rotated by 180 degrees. Put another way, the rotated quarter is upside-down—in the original coordinate system! The positive y-axis was up at the beginning and down at the end.
I’m Just Weird
This is not the standard way of looking at this problem. Generally one would look at the path that a given point takes when a circular object is rolled. When rolled along a straight line, such a point traces the path of a cycloid. With cycloids, one can have lots of fun like proving that the top of a moving car’s tire moves faster in the horizontal direction than the bottom. Does this sound paradoxical? If so, just remember that the top of the tire is moving forward and the bottom is moving backwards; that has little to do with the problem but does make clear that rotating objects tend to mess with our sense of intuition about the natural world.