23 April 2009

Calculus made awesome




22 April 2009

Constrained tourism

Does there exist a Hamiltonian tour of the graph whose vertices are the 48 contiguous United States and whose edges connect states which border each other? More geographically, can you drive through all of the lower 48 states passing through each exactly once? (From 360.)

Here's one possible solution, although I ignored the constraint implicit in the story that provoked the question, which required a start in Michigan. Note that you have to start or end the tour in Maine since it only borders one other state.

17 April 2009

The Art of the Probable: Literature and Probability

From MIT's Open Course Ware: The Art of the Probable: Literature and Probability. The course readings include both some of the classical mathematical writings about probability (Pascal, Fermat, Leibnitz, Bernoulli, Bayes, Quetelet, etc.) as well as various more "literary" pieces.

Only at MIT...

(Seriously, though, I would have liked to take this class. And one of the readings from the last week is "the Bohr-Einstein dialogue", which you may know refers to whether God does or does not play dice.)

06 April 2009

Global octahedron

The Onion, in its fictional world, is owned by Global Tetrahedron. Their logo is a dodecahedron.

(The title of this post splits the difference.)

James Stewart's house

James Stewart, author of calculus texts, has a $24 million house. It has lots of curved walls. Problem: find their areas or volumes, by integrating.

Simmons Hall, an MIT dorm opened in 2002, has a lot of oddly shaped rooms. (I found this silly, because the curved walls meant wasted space -- but I didn't live there, I just had friends who did, so it didn't bother me too much.) The story goes that the Cambridge fire department had trouble giving them a certificate of occupancy because they couldn't determine the volume of certain rooms and therefore couldn't determine whether they were adequately ventilated.

(Article from the Wall Street Journal; link from Casting Out Nines.)

God and some humans play dice

From Tierney Lab at the New York Times:A puzzle in which God, Einstein, and Oppenheimer play dice, and its solution.

04 April 2009

Which two states are closest together?

Fix two sets X and Y in the plane, each the interior of a curve, such that their closures don't intersect. How would one go about finding points x in X and y in Y such that the distance between X and Y is minimal?

Furthermore, say we have a bunch of sets X1, ..., Xn, each the interior of a curve, with nonintersecting closures. Let d(i,j) be the minimal distance between Xi and Xj. How can we find the minimal nonzero d(i,j), that is, the minimal distance between any two sets with nonintersecting closures? (In particular, there should be a faster algorithm than computing all the d(i,j). I suspect O(n log n) of the d(i,j) need to be computed although I have no idea why I'm saying this.)

The problem that inspired this is the following geographic one. What two states in the US that don't border each other are closest together? I think I know the answer; I'll post about that later.