Friday, November 9, 2007

Suppose you have an infinite checkerboard...

Math can sometimes be frustrating, it is true. It will sometime take hours of playing with equations and theorems and such to crack through and understand how to solve a problem. Occasionally when you solve it the response is "oh, that was dumb" - those problems are annoying. Sometimes the response is "huh, thats cute" or "OH!, thats clever." - those problems are satisfying to solve. Sometimes the response is "Wow...that's...impressive..." which usually happens when the solution really is quite long, complex and/or involved - those problems leave you with a sense of accomplishment. Occasionally you find a particularly beautiful solution to a problem and the response is one of pure joy and excitement. The Conjecture and Proof club problem was one of that last category this week. It was amazing. The only problem was that I was then on such a math high that it was impossible for me to go to sleep at a reasonable hour (though I was impressively productive during that time).

Geometric Graph Theory continues to make me thankful for the Core physics required at Mudd. One of our HW problems on the last set was basically a simple mechanics problem, but really messy if you were not comfortable dealing with balancing torques and forces. We are currently playing with double circuit embeddings which result in chalkboard pictures in at least 4 colors of chalk where it is still hard to keep track of what everything is. It is a great deal of fun.

We have started projective geometry in the topics in Geometry class. It is fun to play with things that can't quite be properly visualized. It was also very interesting to see two different models - one completely algebraic in nature and the other based on adding 'ideal' points to regular Euclidean space (where 'normal' high school geometry occurs). On the surface they look completely different, but we showed that they really are the same. During the 'office hour' portion of class today I got him to give us the formal definition of homeomorphic- a notion that I vaguely understood just from the contexts where I had seen it used but which I had never seen defined. It was quite interesting.

I love set theory. The axiom of choice is very strange. It seems intuitive that it should work and there are things that are equivalent that seem very natural, but it leads to some very strange things. The idea that it is possible to put an ordering on the real numbers so that there is a 'smallest' one and any subset has a 'smallest' element seems very counter intuitive (granted, such an ordering is impossible to describe) .


Brian said...

If it's the problem I think it is, the checkerboard one is a very cute problem.

Eric said...

There's nothing counterintuitive about the axiom of choice! The well-ordering principle is intuitively obvious. You just start counting...and keep counting...and keeping counting and counting and counting until you finish! I mean, how is a well-ordering of the reals any less intuitive than one of the rationals? The only difference is that you don't know how to explicitly construct one on the reals.