Thursday, November 20, 2008

My Favorite Math Movies

Here are my top five math movies, along with my favorite quotes. I will not talk about the storylines, because I don't want to spoil the fun.

N is a number : I was sad that I did not see Paul Erdös in person, until I saw this documentary. This is a documentary about Paul Erdös, the wandering mathematician, the man who loved only numbers. Watch this if you want to see him walk, talk, crack math jokes and inspire people around him. You can also see other famous mathematicans collaborating with him.
  • Euler, when he died, he simply collapsed and said "I am finished". An when I told this story, somebody callously remarked : "Another conjecture of Euler's was proven".
  • What is the purpose of life ? "Prove and conjecture and keep the SF's score low".
Pi : I should warn you that this is a disturbing move. Here are couple of quotes.
  • Restate my assumptions: One, Mathematics is the language of nature. Two, Everything around us can be represented and understood through numbers. Three: If you graph the numbers of any system, patterns emerge. Therefore, there are patterns everywhere in nature. Evidence: The cycling of disease epidemics;the wax and wane of caribou populations; sun spot cycles; the rise and fall of the Nile.
  • You want to find the number 216 in the world, you will be able to find it everywhere. 216 steps from a mere street corner to your front door. 216 seconds you spend riding on the elevator. When your mind becomes obsessed with anything, you will filter everything else out and find that thing everywhere.
Proof : This is a perfect math movie with top-notch performances from Gwyneth Paltrow and Anthony Hopkins, an awesome screenplay and a satisfying ending. The following conversation about Hardy-Ramanujan number is aptly placed in the movie.
Robert: Catherine, if every day you lost were a year, it would be a very interesting freaking number.
Catherine: 33 and a quarter years is not interesting.
Robert: Stop it, you know exactly what I mean.
Catherine: 1729 weeks.
Robert: 1729, great number. The smallest number expressed ...
Robert and Catherine: ... as the sum of two cubes in two different ways.
Robert: 123 + 13 = 1729.
Catherine: And 103 + 93. Yes, we've got it, thank you.
Robert: You see? Even your depression is mathematical.
A Beautiful Mind : Well, everybody knows about this oscar-winning movie.
  • Sometimes, our expectations are betrayed by the numbers.
  • Good morning, eager young minds.
  • Find a truly original idea. It is the only way I will ever distinguish myself. It is the only way I will ever matter.
  • [Sol] John, you should go easy. There are other things besides work. [Nash] : What are they ?

Good Will Hunting : In my previous post I mentioned an elementary graph theory problem from this movie. Also, there is a scene in which the characters talk about a math problem and joining two vertices of a tree. Are they talking about 1-trees ? If you don't know what a 1-tree is, read Held-Karp relaxation of Traveling Saleman Problem. The best scenes of the movie are the conversations between Will and Sean.
  • You really hypnotised me, you know.
  • I'm pumped! Let the healing begin!
  • I can't learn anything from you, I can't read in some freaking book. Unless you want to talk about you, who you are.
  • [Will] You just cash in your chips and you walk away ? [Sean] Hey, at least I played a hand.

Btw, Pi, Proof and A Beautiful Mind have excellent soundtracks too. I would like to see more movies of this kind. If you know of any good math movies, leave a comment.

Sunday, November 09, 2008

The Good Will Hunting Problem

Here is a problem from the movie Good Will Hunting, shown in the screenshot below.



For the the graph G(V,E) shown above, find the following :

  • The adjacency matrix A :
  • The matrix giving the number of 3 step walks in G : [Ak]ij is the number of paths of length k from i to j. So, the answer is A3.
  • The generating function for walks from point i to j : The generating function is as follows. Here are more examples of generating functions.
  • The generating function for walks from points 1 to 3 : Simplify the above formula using cramer's rule for i=1 and j=3.