one-dimensional billiards Seeing as school starts week after next, I should probably start blogging more seriously about math-related things, particularly since Pharyngula lists this as an academic blog.
I was reading the latest AMS Notices, and there's an article on kissing numbers by Pfender and Ziegler that's particularly interesting. The kissing number problem in two dimensions is simple: take a quarter, put it on a table, and see how many other quarters you can get to touch ("kiss") your original quarter by arranging them all on the table. It's straightforward to show that the answer is six--they'll be arranged around the central coin in a hexagon. (It's important that all the coins be the same size, so mixing in dimes and pennies is a no-no.)
In three dimensions, you take spheres instead of discs; think of billiard balls. You want to cluster the billiard balls all around the central ball, not just in two dimensions, but in three. It's not obvious, but the maximum number you can fit is 12; this will take on a roughly icosahedral shape. There'll be a lot of extra space, so it was sometimes thought that it might be possible to fit a 13th ball before Schutte and van der Warden proved otherwise in 1953.
In general, the kissing problem in n dimensions is: given an n-dimensional sphere S of radius r, how many other nonintersecting spheres of radius r can touch S at exactly one point? Just as you can fit more around the central ball in 3 dimensions than you could with the discs in 2 dimensions, the kissing number rises with the number of dimensions you're allowed to fill.
So why is this sort of thing interesting? Certainly the people who study the problem are interested in it for its own sake; there's something satisfying about taking a question you might have talked about on the playground and answering it decisively. It's also kind of cool to play with things in arbitrarily many dimensions, even though the novelty wears off eventually. But the real attraction, from a mathematical perspective, is seeing what kind of proof techniques turn out to be useful.
Higher mathematics is all about proving things. (I'm in pure math, so this is my bias showing; applied mathematicians get to worry about other things.) This fact is usually obscured in undergraduate curricula by the first year of calculus, which is impossible to follow from a proof-centered perspective without a foundation of real analysis (here "real" is an appositive noun, not an adjective; "real analysis" denotes analysis applied to the real numbers). Although the official pure math curriculum centers around "mathematical objects" of various kinds, the implicit curriculum consists of proof methods. As such, someone who could recite and even apply a large collection of mathematical results without knowing how to prove anything wouldn't have the foggiest idea what mathematicians do.
Research methodology in other disciplines can usually be explained to people in other fields, but with mathematics this is a tricky business, since developing the methodology for solving your problem often is your research. Mathematics is unique in that the aim of a dissertation is to craft or discover some proposition that, following your work proving it, will be accepted as true without controversy for the entirety of human history. (And also to publish something nontrivial and relevant that will get you a good position and other such banal life-concerns.) Inventing new methods to prove things is an intensely creative process. And it's never clear where a new insight will arise. It might be something so obscure that it's not suitable for blogging, or it might come as a consequence of discovering something as easily describable as the kissing number for dimension n.
Incidentally, from the paper, the kissing number is known for dimensions 1, 2, 3, 4, 8, and 24, with kissing numbers 2, 6, 12, 24, 2480 and 196560, respectively. That leaves a lot of ground to cover.

hello gentlemen,

hello gentlemen,

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well all i can say is that i'm an easy-going girl up for anything, whether it be a night of dancing( shake my booty), a day of enjoying in the sun (its that time of year to work on my tan), catching a plane to a new and exciting destination (no where in particular) would be fun too, or just sitting on the couch catching up on my fav shows (#1 CSI the original) ....but the best is just spending time with friends and laughing like there's no tomorrow. really whatever or whoever can make me giggle is enough.

what i'm looking for in guys??? well, anything goes, because everyone is different in there own way and i don't think i can really describe what i'm looking for because i'm not even sure myself. if you can make me laugh, and show me a good time, i think that's all that matters in my books.

please feel free to drop me a smile...its always nice to receive one....it actually makes me smile :) especially when it comes with a picture...hint, hint...its only fair right?! hehehe

cheers