5/14/2006

squaring the circle (indications of a proof)

"Your daddy he's an outlaw
And a wanderer by trade
He'll teach you how to pick and choose
And how to throw the blade.
He oversees his kingdom
So no stranger does intrude
"

"draw me a square with the same area as the given circle, using a finite compass and straightedge construction. maybe then, i will let you have my daughter."

what kind of twisted challenge is this! even in gongfu movies the most he has to do is to draw a circle and square simultaneously with different hands."

..."a hopeless, meaningless or vain undertaking"

i thought i would save this post for tonight, but bringing it forward a bit doesn't hurt. first up, hello all. been trying to preserve my aura of mystique and conscientiousness by pretending to be studying hard. but here it is. one of my first major exams is here, finally, tomorrow. how i have grown! from a boy who didn't know what exams were when he first entered primary school, wilfully swopping places with another kid just so he could read what he had written for his picture composition, to colouring those same chronological picture boxes from finishing early, through psle, through o levels, through a levels, through army exams, and here i am again.

i always remember a distinct quiet feeling of unease. maybe not stress, i'm too relaxed for that. before the first exam i failed, i remember walking through the ri astroturf clutching frantically at my notes, wishing it would somehow make up for all the work i had not put in through the year. i remember running through the consequences in my mind, vaguely sickened, but i knew that after the exam i left relieved anyway living dangerously as ever until the results came out, at which point i had to face the music.

i've learnt not to take exams so cavalierly since, but there's this bit in me that gets saturated and just takes the foot of the pedal. i am definitely better prepared this time, but there's this bti in me that's still quietly worried. the bit that looks around at other people and wishes he could concentrate as hard as they could (but oh the moments of distraction they have which ones doesn't see!) the fact that it seems somehow, that all the year's work boils down to this, 3 hours, gives this little written exam importance it doesn't deserve. i wish i could go around screaming to everyone, my parents, my friends, to say "look at how much i've enjoyed myself this year, how i've tried to enjoy the course, work hard." and maybe it will suffice. if you want to freak yourself excessively think about how much money is riding on this (the scholarship), how this would severely mess up your future if it doesn't work out, how you will be an unloved child if you don't do well...

maybe it isn't. i guess, at the very least, there will be someone to go for a cup of hot chocolate for. my maths and physics results have always been notoriously volatile. i remember times when i know how everything clicks together and it shows. it seems to be like that a bit now with me and linear algebra (and if i had more options maybe i would take more of that). math is a religion from the axioms. i realise why myself, and so many people around the world, just don't get mathematics sometimes. alright, maybe i am more prone to the occasional calculation error or mental block, but that was never my real problem with mathematics. if i could pin it down, it was because in math you had to be absolutely sure (or at least i felt that that needed to be the case). something follows from everything prior to it, so a wrong step, error carry forward aside, destroys the chain of implications. and those days, when i knew nothing, when everything was to be "just memorised", i had no idea what steps i was taking and i was groping in the dark really. but now maybe i am better at taking the leap of faith, learning that some things are taken as given, i shouldn't analyse them excessively, and then go along with whatever is required to solve the question.

exams aside, but still on geometry. shel silverstein once wrote a famously endearing tale "the missing piece", simply illustrated with an incomplete circle looking for its wedge. i must show it you you sometime if you haven't read it. it is so hard to make a circle happy. and i just thought of this in the shower. do you know what a selfish creature the circle (or sphere, or higher dimension analogues of it) is? think about it. take a square, and try to fill the paper. you will probably do it fine. even a triangle, you could use it to cover the same space. now take a circle. try to use it to fill space, volume, and if you calculate all the area left behind in between circles, i am sure it is larger than most other well-behaved geometric objects. you could even try to prove it.

and it's funny because the circle is always the shape i have identified for people (well who likes to be called a square?) i remember the well worn phrase (the right for you to swing your arm stops at another person's face). that's just how we are then, circles with our own personal space, our circumscribed radius. keeping all this space in between.

just as said circle needed a wedge, there's someone from time to time who will meld or change the contours of our existence. (a use of the word contour that doesn't come in the same sentence as "lagrange multiplier". yay!) to give an edge to the circle, to make it more of a polygon, take up less space, nudge other polygons in the side with its jagged corners. yes, there's still something beautiful about the perfection of the circle, but ideals are hard to live to , trying to maintain our vain perfection everyday. we are rarely a polygon whose sides tend to infinity but more often than not a hodgepodge collection of lines.

where these lines meet. (is an intersection). skew lines on the other hand are the most tragic. heading for a collision course, but yet always destined to be on a different plane. at least parallel lines walk in tandem with you. god must have been a geometer. and the contours of a body, a hand here, a face there, all seem surprisingly natural and easy. occasional incongruity aside, but i love incongruity! i love the roses hanging upside down withered and dried off the clothespins, still retaining their green thorned stems and with the dye fading off the white rose. yes i was swindled, in more ways than one.

3.14159265457979...
transcedental.
i will teach you my mmemonic for the number pi=p

http://mathworld.wolfram.com/CirclePacking.html (i found something about it).
i just realised the smaller the circles get, the better they pack, until they are infinitely small.

err yah this is like me before my 2nd driving test (one of the most stressful tests in my life)
what if i fail again? haha. take the ipod and walk around listening to music. NEVER NEVER arrive early for an exam.

3 comments:

mich said...

i'm sure i'm gonna forget to ask you so i'm writing it here

who are the skew lines (random thought)? how were you swindled in the second way?

chanlek said...

Do you always have to write like this n make others feel like they are unworthy of holding a pen?

EƤrendil said...

"math is a religion from the axioms" what a wonderful quote!
G-d bless mathematics!
omg i am so math-sick that i actually enjoyed reading it although i expected a formal proof of that ancient problem!
hows everything gg in london? is it that stressful the summer school or is it a bit relaxing? what courses r u taking? did u find a house for next year? who r u gg to live with? where?
okkk...have fun! dont study so hard!
Miguel