...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...

Wednesday, May 15, 2002

OK, so I'm stuck on a problem tonight (again), not unlike a moth that wandered into an attractive shoebox, and can't get out (moths are pretty stupid, if you'll recall). Now I'm buzzing around inside the shoebox, hitting its walls at random times with surprisingly solid thumps. That's figurative, of course, for all i readers out there - I don't normally buzz, as such, and I rarely hit walls (that only tends to happen in the dark, or when I'm distracted by, say, attractive human females or some particularly interesting representative of the local fauna ... for quite different reasons, I assure you!) Though I'd have to admit on cross that I actually might be said to 'buzz' after a solid dose of some hypergolic substance, such as Mexican food. Said buzz means it's generally time for any bystanders to head for the shelters, by the way. But that's getting rather off-topic (good thing I caught myself before I ended up talking about the hot and heavy relationship between a certain comely Floridian manatee and Donald Rumsfeld in a third-degree parenthetical). Which is amusing, because I haven't actually defined any topic yet. My rambling talents are impressive, n'est pas?

So, you i readers are probably wondering what problem it is that has me thinking about being a moth. It's a simple-sounding one, really. For future reference, the problem in question is Exercise #10 in the Chapter of Pleasure Spiked With a Considerable Amount of Pain<, a.k.a. Chapter 3: Mobius Transformations And Inversion. So. I'm being asked to prove the following: Any two non-concentric non-intersecting circles can be mapped to concentric circles by means of a suitable Mobius transformation. (From here on out, I'll refer to a Mobius Transformation as M(z), to save typing and make things more cryptic, and also to save me the headache of trying to figure out how to get a double-dot above the 'o' in Mobius, which ought to be there.)

Now, as the first step of the proof, it's suggested I consider the following: Grab two circles, A and B, as specified above, by the lapels, and show that there exist two points that are symmetrical with respect to both A and B. Now, the first time I sketched this, while sitting on the Throne Of Power, this seemed as helpful as single-ply TP to a guy with hemmorhoids and a spelling problem. Further reflection (read: taking care of nature's business, which I promise not to discuss further here) allowed me to come up with something interesting. Draw a line through the centers of A and B. If these mystical symmetrical points exist, they damn well BETTER be on this line, right? I mean, it just makes sense, if you squint at it enough!

Now, squint some more, and don't forget to blink thoughtfully while furrowing your brow. It'll start to make a hazy kind of sense that one of the points ought to be 'close' to the center of the inner circle, which means that the symmetric point ought to be 'far away' from the small circle's circumference. But if you get the position of the point right, the symmetrical point ought to be so 'far' away that it'll be beyond the circumference of the outer circle. Fiddle with the position of the point on the line a bit, and you should be able to get it so that the image point is symmetrical with respect to both the inner and outer circles. Or so the voices in my head tell me.

The above, for the uninitiated, is known as a handwaving argument, due to the intellectual similarity of the user of said argument to a lizard that just found out from a book that it shares an ancestor with birds. It kinda-sorta makes sense, but it's quite far from an actual proof, or anything concrete and analytical and mathematical. How to actually get from the "I think I'm a turkey, I can fly if just flap hard enough!" stage to the "Eureka! Here's the solution" stage is too complex a process to fit inside this margin sometimes referred to as teh Intarweb.


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