Prove that at any time there are two opposite points along the Equator, which have exactly the same temperature. Assume the temperature function varies continuously as you move along the Equator.

Counterargument: This is patently impossible. If there are such points on the Equator, there must also be similar points on any circle around the Earth, such as a meridian. But in that case, we'd have one point in the north hemisphere, in winter, and the other in the south, in summer; that doesn't make sense!

What's wrong with this reasoning?

As one travels around any closed path on the surface of the earth, every temperature will be recorded at least twice since you start at temperature A and you have to get back to it.

The angular distance between two points of equal temperatures varies contiuously between 0 and 2pi, so in particular there are at least two points separated by an angular distance pi.

The counter-argument is not a counter argument; the points exist and simply fall close to the equator. This can be seen by drawing a simple plot which is cold on the northern hemisphere and cold in the southern one.

Posted by ajosin on 2005-03-28 16:14:58