## Meteorology and Ham sandwiches

April 5, 2004

I'm putting together a presentation for my topology class on the Borsuk-Ulam theorem. Its a theorem about what happens when you have a function from a sphere to a plane. I'll omit the details here, but the theorem has several amusing correlaries. First, at every point in time, there are at least two points on the Earth which are exactly opposite each other that have the same temperature and humidity. Think about that for a second. That spot could be your house. Somewhere in India or China, someone might be experiencing the exact some environmental conditions as you. Second, suppose you wanted to make a ham sandwich consisting of 2 slices of bread and one slice of ham. Then no matter where the bread and ham are, you can always slice them each exactly in half with only one cut of a knife (if your knife was infinitely long). This means that even if the bread is still in the bag and the ham is in the refrigerator, you could cut the sandwich before making it, and still only use one stroke. Crazy stuff.