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stoutfellowOne of my old math professors said that there were two kinds of mathematicians: the simple-minded, and the woolly-headed. Simple-minded mathematicians make their way through the forest tree by tree; they worry at little details, and bull their way through. Their particular fault is a failure to see the big picture. Woolly-headed mathematicians see the whole forest and the general shape of the path, but sometimes overlook the details. (He was one of the woolly-headed ones; in topology class, his theorems usually began, "Let X be a nice space." What "nice" meant, precisely, he never wanted to specify, though if pressed he'd spit out "semi-locally simply connected weakly Hausdorff space". I once knew what that meant....)
Anyway, I'm one of the simple-minded. When I work on a problem, I'll pull in every random tool that comes to hand and just beat it to pieces. The difficulty comes with the write-up: the first drafts of my proofs are inevitably Rube Goldbergesque, with bits sticking out in all directions. The major work is trying to pretty it up.
I'm at that stage with the prisms paper, currently writing up Case II. Here's how Case II works. A first pass shows that there are fourteen possible situations, and the task is to show that none of them is actually possible. There's a simple trick that shows the fourteen are "really" just seven. (Simple, yes, but it gave me fits for a while. There are two ways to do the computation, and I kept getting different answers from them: one said "eight", and the other said "six". Eventually I saw what I was doing wrong in both, and settled on seven.) Next, there's another easy trick that shows that three of the seven are impossible. Four to go.
Okay. These two, there's a simple geometric argument that knocks them out. This one... okay, here's an obviously-true-but-hard-to-formulate-precisely argument, with faint overtones of calculus. The last one... yuck. Here's an intricate counting argument that settles it. (I kept losing track of how that last one goes, and trying to write it up was a serious pain.)
This is no good. I've got three arguments of completely different flavors, trying to prove a single theorem. The last couple of days, I've been trying to come up with a single framework that captures all four remaining situations. It's been slowly coming into focus; one of the ideas I use heavily in Case III can be deployed here as well (always a plus to re-use stuff!). Coming up with the right generality... If I don't insist on this being a piece of a geodesic, I can apply it more easily in Case III too. So what is it a piece of? A disjoint union of parallel linear cycles? Ugly, ugly.... Wait a minute, this thing doesn't even have to be a face of the prism; all it needs to be is a random rectangle. But then I don't need to mention geodesics at all here. That means this whole lump of stuff should be pushed up to the beginning of the section. Now the counting argument suddenly looks a whole lot simpler, and the calculus-flavored thingy can be split in two, one to go early and the other to be an application of the same ideas as in the counting argument. YES!
Being at home, not having access to any of my special software (Scientific Workplace, Mathematica, Geometer's Sketchpad), throws me back on my own resources. I think that's a good thing, and the reason why this is coming clearer now than it did during the workweek. I'm giving myself until the end of April to finish this thing off.
Anyway, I'm one of the simple-minded. When I work on a problem, I'll pull in every random tool that comes to hand and just beat it to pieces. The difficulty comes with the write-up: the first drafts of my proofs are inevitably Rube Goldbergesque, with bits sticking out in all directions. The major work is trying to pretty it up.
I'm at that stage with the prisms paper, currently writing up Case II. Here's how Case II works. A first pass shows that there are fourteen possible situations, and the task is to show that none of them is actually possible. There's a simple trick that shows the fourteen are "really" just seven. (Simple, yes, but it gave me fits for a while. There are two ways to do the computation, and I kept getting different answers from them: one said "eight", and the other said "six". Eventually I saw what I was doing wrong in both, and settled on seven.) Next, there's another easy trick that shows that three of the seven are impossible. Four to go.
Okay. These two, there's a simple geometric argument that knocks them out. This one... okay, here's an obviously-true-but-hard-to-formulate-precisely argument, with faint overtones of calculus. The last one... yuck. Here's an intricate counting argument that settles it. (I kept losing track of how that last one goes, and trying to write it up was a serious pain.)
This is no good. I've got three arguments of completely different flavors, trying to prove a single theorem. The last couple of days, I've been trying to come up with a single framework that captures all four remaining situations. It's been slowly coming into focus; one of the ideas I use heavily in Case III can be deployed here as well (always a plus to re-use stuff!). Coming up with the right generality... If I don't insist on this being a piece of a geodesic, I can apply it more easily in Case III too. So what is it a piece of? A disjoint union of parallel linear cycles? Ugly, ugly.... Wait a minute, this thing doesn't even have to be a face of the prism; all it needs to be is a random rectangle. But then I don't need to mention geodesics at all here. That means this whole lump of stuff should be pushed up to the beginning of the section. Now the counting argument suddenly looks a whole lot simpler, and the calculus-flavored thingy can be split in two, one to go early and the other to be an application of the same ideas as in the counting argument. YES!
Being at home, not having access to any of my special software (Scientific Workplace, Mathematica, Geometer's Sketchpad), throws me back on my own resources. I think that's a good thing, and the reason why this is coming clearer now than it did during the workweek. I'm giving myself until the end of April to finish this thing off.
