...Prove Their Worth...

"Problems worthy of attack
prove their worth
by hitting back." - Piet Hein

A kind of running diary and rambling pieces on my struggles with assorted books, classes, and other things, as they happen. You must be pretty bored to be reading this...

Friday, June 07, 2002

I do my best work sitting on the shitter. I swear. Today was another example. I've been struggling since yesterday with a very interesting problem from Ch. 4 of Needham, #18, I think. It has to do with the following interesting question. Say you've got two curves touching (gently, non-pathologically) at a point. Is it possible to define a meaningful 'angle' between these two curves at this point? Well, maybe. Certainly, any such 'angle' that we end up defining better end up being conformal. That is, if we apply a conformal transformation to the plane containing the two curves, it will preserve any normal angles, like those that are part of triangles, say. ('Conformal' is just a fancy way of saying 'angle-preserving'. An important bit of trivia is that analytic transformations are always conformal.) So our made-up definition for the meeting "angle" of two curves damn well better act like normal angles do, or it doesn't deserve to be called an 'angle'.


This actually turns out to be a fairly hard problem, and it is only fully solved in the last chapter of Needham's book. In chapter 4, though, I'm asked to follow a couple of doomed attempts at a definition. The first attempt at a definition was done by Newton. Unfortunately, with the modern tools of complex analysis, it is fairly simple to show that it is not conformal. So it doesn't work. There is another attempt to define this 'angle', building on Newton's attempt. It fails too, but for a different reason. (As I said, this won't be solved until like chapter 12 or 14 or whatever.) But never mind that. What's important is that last night I got stuck on a bit of geometry in showing that this second attempt doesn't quite work. I've been thinking about it all day, to no great benefit.


About an hour ago, while resting on the porcelain throne in preparation for a nice, fat-arse shaking jog, the answer to my quandary hit me like a lightning bolt from a minor Greek god, or more poetically like a toilet-alligator bite on the ass. See, there's an often used fact that for small theta, sin(theta) = theta. And that is the lever that got me unstuck.


I wonder if it's unusual for people to do their best thinking while sitting on the crapper, and if it isn't, how many of the great discoveries throughout history were made while taking a dump? Perhaps Archimedes wasn't in the bathtub when he bellowed his now-famous cry, "Eureka!"


Here's a poetic version of the above post, mostly due to underthumb's help:


Lo!
Though I poop,
My mind doth not droop,
And from it--for sooth!
Ideas now go 'gloop'!

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