Frank Morgan's Math Chat
March 18, 1999
"If someone told us that on a distant planet there were blue horses, we could believe it. But if they said that three plus four horses equaled ninety-seven horses, we would know it was impossible. . . . Bertrand Russel says that it is an error to speak of the evolution of mathematics. As soon you say that three plus four equals seven, that statement implies all of calculus, algebra, etc. On the other hand, when you have discovered the wolf, you have not discovered the kangaroo or the cat. . . . To illustrate impossibility, the Romans called something 'a black swan.' Well it turned out that in Australia, there were nothing but black swans."
OLD CHALLENGE. Which countries in the world have a point such that the shortest line from every other point in the country stays inside the country? (Pretend that the world is round and smooth: ignore mountains and valleys.) Mathematicians call such countries starlike. Are there any countries such that the shortest line between any two points stays in the country? Mathematicians call such countries convex.
ANSWER. Most countries have very irregular sections of boundary, including rivers or coastlines, which prevent them from being starlike or convex. If you include territorial waters, the South Pacific island country of Nauru, for example, seems to be a starlike "rectangle," bounded by lines of latitude and longitude. It may look convex too, but it is not: the shortest path between two points along the southern border follows not the boundary (line of latitude) but the shorter "great circle route" which passes south of the boundary; this works because the distance around the earth in the southern hemisphere gets shorter as you get closer to the south pole.
NEW CHALLENGE.It is well known that the circle provides the least-perimeter way to enclose given area in the plane. What is the least-perimeter way to enclose given area in the surface of the unit cube? (What about the hypercube in four-dimensional space?)
JOKE. This week's winning math joke comes from David Shay and Al Zimmermann:
Noah's ark lands after The Flood. Noah lets all the animals out and says, "Go forth and multiply."
A few months later, Noah decides to take a stroll and see how the animals are doing. Everyone has lots of babies except for one pair of little snakes. "What's the problem?" says Noah.
"Cut down some trees and let us live there," say the snakes. Noah does as they ask.
A few weeks later, Noah checks on the snakes again. This time there are lots of baby snakes and everybody is happy. Noah asks, "Want to tell me how the trees helped?"
"Certainly", say the snakes. "We're adders, and we need logs to multiply."
Editor's note: Of course this was long before calculators, when some folks used to multiply numbers by adding their logarithms.
Send answers, comments, and new questions by email to:
Frank.Morgan@williams.edu, to be eligible for Flatland and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan's homepage is at www.williams.edu/Mathematics/fmorgan.
Copyright 1999, Frank Morgan.