...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...

Saturday, May 18, 2002

Hrm. Well, I'm still stuck on that particular problem, and it's gotten frustrating enough that I spend about twenty minutes on it in the evenings, then move on to something else, lest I go more nuts with frustration, and assault any rabbits and/or cats I run into at night while jogging with warbling [ala Morissette] romantic serenades to cactuses.


The something else in this case means rereading Chapter 3. Currently, I'm reading about the visualization of a generalized M(z). It turns out to be quite an interesting matter, with lots of nice pictures involved - various whorls and curves and so on. I saw that it was interesting the first time through, but now I find that I'm following it a lot better (perhaps it helps having read the chapter on complex differentiation and analytic functions and so on, which comes later).


In true form for myself, I've of course run into problems with the material. Amusingly, it's not with the core of the material as such. As part of the setup-attack, Needham juggles various constructions and transforms and compositions of transforms, and uses 'and clearly' several times. Somehow, I always get bogged down when people start farting around with rapid combinations and discombinations of transforms. If I go slow, furrow my brow, and carefully and deliberately plow through, not forgetting to alliterate compulsively, I follow things, but if i just 'read', I don't. :-( Anyways, onto things I do follow.


It makes sense to me that you can pull a 'multiplier' out of Mobius transform (it's not unlike pulling a tooth, but it's not like pulling a tooth, either, as it's not fundamentally and irrevoccably unpleasant, and the weaponry of basic linear algebra* [i.e., squeezing an eigenvalue out of the matrix representation of the transform] acts as an anaestetic.]) I'd even puff my chin out enough to say that I grok how the nature of that multiplier determines the 'spirit' of the transform - if it's a dilation, the transform is elliptic (curves concentrating around the fixed points of the transform), if it's a rotation the transform is hyperbolic (curves from one fixed point to another), and if the multiplier is a combo of a rotation and a dilation, the transform is loxodromic (looks like cool whorling curves from one fixed point to the other). The first two types of transform, it's neat to note, look a whole hell of a lot like EM field lines, and the loxodromic one might look kind of like the field of one EM 'pole' moving past another (?). Who wants to bet there's a major application here, and they actually are EM field lines?


Here's the picture I've got in my head right now. It's probably completely wrong and insane, but so it goes. A Mobius transform can be looked at a linear transform on C2 acting on homogenous coordinates in C2 of a point z, which lives in C (note: homogenous coordinates just means you label a point z by a ratio of two complex numbers, which are then your 'homogenous coordinates'. Sounds pedantic, confused, and fancy, but it's all made clear on p. 158) That reminds me of the obscene volumes of delicious Japanese food I consumed today at the Hinode during the 10 buck/person lunch-time buffet. Mmm. But moving on...


Now, if you want to avoid headaches (a good thing in general if one isn't Zeus - not everyone can get someone as cool as Athena out of a headache), you'll deal with Maxwell's equations in an explicitly relativistic mindset - with four-vectors and Fuv and all that shit. I think. I.e., you're now playing on Minkowski's playground, and need to deal with four dimensions.

Now, I also remember reading a snippet about spinors somewhere - they're column vectors of two complex numbers. Now, I know Penrose came up with them, and I know for a FACT Mobius transforms have a lot to do with relativity (the book says so, and refers me to Penrose's work on the subject - I looked at it, but it's too hard-core for me. I'm still on the soft-core diet, and I'm not ready for the 'German scheisse video' (in South Park lingo) of math books. So, apparently, you can do EM with spinors, and there are advantages to this. Now, what I'm ponderously siddling up to is that you can pick up this 'homogenous coordinate' shit by its collar, give it a good shake and poke it with a sharp stick in its tender spots, until it admits it's actually a spinor, and then do EM from this perspective. Which is why pictures of stuff like Mobius transforms ends up looking suspiciously like EM field lines.


Note to the gullible (i.e., self): This is quite likely to be amusingly wrong, so don't actually bet on it! But it'd be funny if it was even remotely close to the truth.


In other news, I have a new watch, one that I can actually read, due to the face not being so scratched up as to raise questions of whether I tame almost wild grizzly bears in a circus for fun and profit. And it's pretty, so I'm actually at some risk of getting mugged for it - the previous watch was more likely to shake a would-be mugger into giving me his watch out of pity. Also, I got a book (as a present) called "The Glass Bead Game", by Hermann Hesse. It's quite thick. It won the Nobel Prize in Literature in 1946, and people I trust say it's really, really good. So it's probably really good. Look forward to reading it.


My fingers are tired.



* -- Note to self: a decent online text for linear algebra exists. Read it sometime. You really, really need a refresher. [cough]


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