Cute Math Curiosity

Posted by Graham Wheeler on Tuesday, May 6, 2008

I heard this proof a couple of days back and thought it was fun and elegant. If you draw a circle anywhere on the surface of the earth there will be at least two opposite points that have the same temperature.

The proof relies on Bolzano’s theorem, which is an instance of the Intermediate Value Theorem. Bolzano’s theorem says that if a continuous function defined on an interval is sometimes positive and sometimes negative, it must be 0 at some point.

We assume that temperature on any path on the earth’s surface is continuous. If f(x) is a continuous function, and g(x) is a continuous function, then f(x)-g(x) is also continuous.

So, choose some starting point on the circumference of the circle, and let f(x) be the temperature at the point at offset x clockwise from the starting point, and let g(x) be the temperature at the point opposite that point. If the temperature at some point x is warmer than at its opposite point x’, then f(x)-g(x) is positive, while f(x’)-g(x’) is negative. So, by Bolzano, there is a point where f(x)-g(x)=0, i.e. f(x)=g(x).