"Hm. Let's do something simple - consider the infinite 3-regular graph ."

Mathematicians are weird.

The thing is, you have to wait around to figure out what the things are that make sense to you without thinking, which are the ones you have to take a few seconds to figure out, and which are the ones that feel beyond you when you first encounter them, but really you need an hour with a blank piece of paper with an afterthought of follow-up questions. For example, something "simple" today had me simply reeling as I couldn't get it into my head (having to do with matrix multiplication, which is something I learned in high school - though not in such an abstract way) and then, later in the day (at the colloquium talk I just left) something that came to my mind without thinking: picture a lattice, like the kind on top of a pie. Basically, looks like a checker board or chess board - vertical and horizontal lines with squares in between. Now, there can be other kinds of lattices - ones that don't have little squares, but rather, little triangles, or parallelograms, or -- hexagons. In the talk today (about percolation - the way liquids move through particles, like how far water gets if it has to seep through a bunch of densely-packed sand), it turns out that in percolation theory, many results can be proved when using hexagonal lattices but not with the traditional square ones. I asked why. "Because," the speaker responded, "The faces of the hexagons are the vertices of triangles." And I had the picture in my mind immediately.  (Grab a piece of paper - draw a dot on the center of each hexagon you draw, and connect them - you'll see what I mean. This result is not all that deep or anything - I was just happy to visualize it so quickly.)

Okay, okay, so the more I think about it, the more pathetic a victory that tiny moment was. However, I've been thinking today. Quite frequently, in my head, I think about how I don't feel like I'll ever be worthy of the title 'mathematician' because there are so many people who know so much more about math than I do - so very, very much more. But all of a sudden as I was feeling sorry for myself like this this morning, I thought of a tiny fish saying that it can't swim because it doesn't go as fast or as deep as the mighty narwhal, for example. What a stupid thing to say. Of course it can swim! Damn well, too!

Hold on, though. Maybe, in this metaphor, I might actually be a sea lion- because, I can swim - and in some areas of the ocean, I may even be a force to be reckoned with - but I like to spend a lot of time on land (a.k.a. playing music, reading, thinking about philosophy, psychology, economics, etc) and not in the water (i.e. the murky and seemingly infinite depths of math). In fact, I can't be in either place indefinitely.... Oh dear. Metaphor carried me away. I'll swim back to the point.

So, the point is, as you can clearly see, the GRE is in less than 48 hours and I'm on my last mental legs. But at least, I understand how hexagons can be like triangles. Guess that's a small victory.