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?
The Borsuk-Ulam theorem states that for any continuous function defined over the surface of a sphere, there will always be some point on the surface of the sphere where the value of the function is equal to the value at the point on the opposite side of the sphere.
But this theorem doesn't state that the two matched points will necessarily be on the equator.
Hopefully someone is more familiar with this theorem than I am.
[unless maybe the circle of the equator can be considered a 2 dimensional sphere]
Edited on March 27, 2005, 7:31 pm
Posted by Larry
on 2005-03-27 19:30:02