Saturday, August 22, 2009

von Neumann

I'm reading a biography of Johnny von Neumann by Norman Macrae. Or maybe I should say the biography, it's not like there are many choices for biographies of this great mathematician and early computer scientist. The book is not horrible and it's the only game in town, so I am reading it. It's based on material collected by Stephen White---who was somehow involved with Stan Ulam in an earlier version of the project.

The basic problem with the book is that Macrae is a whack job. To quote Mark Yasuda's review at Amazon:
By far the biggest problem, however, comes from MacRae's approach to the book - he insists upon inserting so much of his own world views and dogma into the body of the book, that we no longer have a biography on Von Neumann - we have Von Neumann's life used as a vehicle for MacRae's own personal views on education, politics, the Japanese economy of the 1960's through the 1980's (I never expected to see this in a Von Neumann biography), and cold war history. He takes time out to provide slanted views of Bertrand Russell and Norbert Wiener, for no reason (they barely figure in the book beyond his distorted descriptions of them) other than to insinuate that their liberal viewpoints are due to poor parenting. In sum, the book's most fatal flaw is that there's entirely too much of MacRae, and not enough Von Neumann.

Leaving all that aside, the book contains yet another retelling of the famous story about von Neumann and his approach to this problem:
Two trains (or bicycles) are 20 miles apart and headed directly toward each other at a speed of 10 mph. A fly starts from the front of the train on the left, flies at a speed of 15 mph to the tip of the train on the right, then turns around immediately and flies back to the first. This cycle continues until the trains meet, ending everything. How far does the fly fly?

The story is that when posed this problem, von Neumann thought for a second and gave the answer (below). When asked how he did it, he answered "I summed the infinite series, of course."

The easy way to do the calculation is to focus on the trains:

1. The trains will meet in one hour.
2. The fly will fly 15 miles in that hour.

What was interesting to me is that, when you think about the series method, it is not that hard, and converges rapidly.

In the first cycle the fly and its target train start with a separation of 20 miles. The fly's trajectory covers the initial distance times the ratio of its velocity to the total velocity of approach (15/(10+15)):

d1 = 20 miles * 0.6 = 12

However, each of the trains also moves 12 miles * 10/15 = 8 miles during the same period, so the new distance for the second cycle is 20 - 2*8 = 4, or one-fifth of the original. If you follow this out you can see that the series we need to sum has its first term equal to 12 and each succeeding term is 1/5 of the preceding one. The first four terms sum to:

12 + 2.4 + 0.48 + 0.096 = 14.976

I think Johnny guessed at this point.

Come to think of it, there is probably a theorem about infinite series that says if we add terms like this, the sum of all the terms obtained from 12 by multiplying by 1/5, that will be equal to 12 * 1/4. Anybody know?

[UPDATE: Not sure if this was laziness (I was in a hurry) or being incredibly dumb. See wikipedia. This is the geometric series with r = 1/5 and a = 12. The sum is a times 1/(1-r) = 12*5/4 = 15. ]

No comments: