Welcome to the 36th Carnival of Mathematics. I think I’m supposed to have something cute to say about the number 36, so we’ll start with something from my personal background. Though I don’t claim to be one of them (or believe in it in any way), according to Jewish tradition, there are 36 good and righteous men whose goodness is all that stands between the world and destruction.

More importantly, 36 is a square number, and squares are the jumping off point: jd2718 asks about polygons whose area and perimeter are equal (well, once you ignore those pesky units). He first shows that you can set up any square to have both be 16, and then poses the problem to his readers. If you have any ideas, go let him know.

Over at catsynth, who hosted the previous carnival, has a post about math cats, which seems to get to some reasonable stuff through the instructional ability of cats. Seriously.

Moving beyond cats, we get to learn about Stacks from A. J. Tolland at the Secret Blogging Seminar. Granted, he doesn’t actually DEFINE what a stack is, but it’s still a good exposition of how they work. Also check out the comments for some stuff on classifying spaces and stacks of negative dimension…I’ll bet you were naive enough to think that anything legitimately geometric had to have positive or zero dimension.

From 0xDE, we get two posts. The first talks about Pythagorean triples and Babylonian mathematics, and at the end mentions a connection to the abc conjecture in number theory, which, among other things, would imply Fermat’s Last Theorem for all exponents greater than 6 (if my notes from number theory as an undergrad are to be trusted). The second post is less number theoretic and more graph theoretic, as it examines cubic symmetric graphs, that is, graphs where each vertex has valence three and the symmetry group is transitive on both vertices and edges. The post talks about embedding them into space and a bit about flat tori.

From Out In Left Field, we have a post detailing how math is taught on the ground in public schools, as opposed to how mathematicians seem to think that it’s taught. I’m baffled by the mathematicians being quoted here, in fact, because I seem to remember math not being abstract ENOUGH for me when I was first learning it. Perhaps it’s just been less time since I was studying at that level than they have.

At Math and Logic Play, we have another simple puzzle, this time focused on probability. How much does what you know affect the probabilities in this situation?

At Neverending Books, we get to see what distinguished mathematician Vladimir Arnol’d though about the frequency of which things occur in threes in mathematics, things like solids, division algebras over the reals, exceptional Lie algebras, and the like.

Isabel of God Plays Dice has given us the following variant of the Travelling Salesman problem, famous for being NP complete. Is it possible, in 26 days, to see a baseball game at every major league park? Well, some guy is trying, and Isabel is there to analyze whether or not he can do it, and how.

At Topological Musings, we get a lesson in basic category theory from Todd Trimble. Todd followed it up with another category theory post later, and generally is trying to help push the categorical framework for thinking about mathematics out there, and that’s something I happen to agree with strongly.

As does John Armstrong, the Unapologetic Mathematician. His post on Sphere eversion takes a rather categorical point of view and describes how people have used it to understand sphere eversions better. In fact, he uses higher category theory, and in the post includes a link to a large pdf which gives step by step images of the eversion itself.

Moving from spheres to discs, Walt at Ars Mathematica gives us a post on the Brouwer Fixed-Point Theorem. He mentions Sperner’s Lemma, which gives a completely combinatorial and constructive proof, and even mentions a little bit about the theorem’s importance in mathematical finance.

Next up is Gil Kalai’s post at Combinatorics and More which starts from Euler’s formula and then gets going, talking about flag numbers and the CD index. The flag numbers seem suspiciously like flags of vector subspaces, so perhaps that’s something I should look into a bit more…

From 360, we get a post titled “The Calculus of Crabbing.” It starts at looking at what oceanic conditions are best for crabbing, and then moves on to calculus, the tides, and how it all fits together.

And finally, a second post form the Secret Blogging Seminar, this time by David Speyer. David wants all of us to be able to write down representations of GL_n. In doing so, he gives a bit of a description of some representation theory, mentions algebraic representations of reductive groups, and quite a few other related things, like the Peter-Weyl theorem.

As far as I know, there isn’t a host yet for this Carnival next time, so someone go volunteer. Anyway, that’s it for now. Here’s a link to the blog carnival page if you want to offer to host, or just submit posts for future Carnivals of Mathematics.