Although I’ve always enjoyed math class, growing up I wasn’t someone who necessarily wanted to be a mathematician or a math teacher. I also wasn’t someone who pictured herself decades later blogging about her favorite math theorem. But here we are.
My favorite math theorem is the Borsuk-Ulam Theorem. Here’s an explanation from Mudd Math Fun Facts:
“The Borsuk-Ulam theorem is another amazing theorem from topology. An informal version of the theorem says that at any given moment on the earth’s surface, there exist 2 antipodal points (on exactly opposite sides of the earth) with the same temperature and barometric pressure!
More formally, it says that any continuous function from an n-sphere to R^n must send a pair of antipodal points to the same point. (So, in the above statement, we are assuming that temperature and barometric pressure are continuous functions.)”
It goes on to explain the simplest iteration of this theorem:
“Show your students the 1-dimensional version: on the equator, there must exist opposite points with the same temperature. Draw a few pictures of possible temperature distributions to convince them that it is true.”
Think about that. There HAS to be two opposite points on the Earth’s equator with the same exact temperature. Woah. Here’s how we know it’s true:
“The one dimensional proof gives some idea why the theorem is true: if you compare opposite points A and B on the equator, suppose A starts out warmer than B. As you move A and B together around the equator, you will move A into B’s original position, and simultaneously B into A’s original position. But by that point A must be cooler than B. So somewhere in between (appealing to continuity) they must have been the same temperature!”
Isn’t that cool?
Citation: Su, Francis E., et al. “Borsuk-Ulam Theorem.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.