eiπ + 1 = 0
Euler's Identity. 'tis a thing of pure beauty.
Three very suspicious numbers in a menage-trois, creating something real. How can two irrational numbers and an imaginary number work together to make a very real integer? It boggles the mind entirely. Somewhere in a past left behind, this was the first equation to make me sit up and consider imaginary numbers as something more than a trick.
The standard wuss way of explaining this (as happened to me) was that of pre-cooked trigonometry.
eiϑ = cos(ϑ) + i sin(ϑ)
But that is just a completely arbitrary equation, when you really think about it. And I'm an incorrigible skeptic. But that's where that lesson ended and Math is not taught as much as lectured on. But somewhere during my engineering, I learnt about the Taylor series, for approximating sine and cosine values. Except, it's not really an approximation, but an infinite series and the partial sum, is used for approximation.
ϑ3 ϑ5 ϑ7
sin(ϑ) = ϑ - — + — - — + ...
3! 5! 7!
ϑ2 ϑ4 ϑ6
cos(ϑ) = 1 - — + — - — + ...
2! 4! 6!
Remember, that works on radians, not regular 'ol degrees. So ironically, when you throw the magic number in there and spend an eternity calculating it, the sin(π) works out to be one huge zero. And it has to, because looking at it from pure geometry and sine as a pure fraction.
Now, I never understood how an infinite number of operations could ever result in a finite number. Well, it's the ghost of something familiar - Zeno's Paradox. And well, Archimedes debunked it, way before I could even attempt it.
Now if you shift a little from geometry of lengths into the world of co-ordinate geometry, you suddenly realize that imaginary math is literally co-ordinate geometry in disguise, except with imaginary numbers (woooo ...). Pull that very Eucledian right triangle into a unit circle on the imaginary plane, the boundaries between the disciplines start to disappear.
The imaginary pixie dust sprinkled on e results in another taylor series expansion, which ironically just shows to go how you can really go mad learning mathematics. Now, the taylor series expansion for just plain 'ol ex goes like this.
x2 x3 x4
ex = 1 + x + — + — + — + ...
2! 3! 4!
Now, here's the clincher. If x just happened to be i, the alternate coefficents would be negative. Oh, yes ... that's pure imaginary pixie dust, but once you get hooked on it, there's no getting off it :)
Now, we get to the final and crucial equation all over again.
eiπ = cos(π) + i sin(&pi); eiπ = -1 + 0 i eiπ + 1 = 0
Time to run out on the streets and yell out that ... "they were right, eiπ is REAL!".
--That's not right! Heck, that's not even wrong!
-- Wolfgang Pauli
Inner beauty is overrated.
Perhaps the greatest handicap my parents ever provided for me was the concept of inner beauty. In their attempts to prevent me from turning into a flake, they emphasized that it was probably the most important thing to develop. Their efforts bore fruit. They taught me to look deep into the heart of others, judge them by their intentions and to know them by their actions.
But they also taught me to avoid the shallow. I learnt that the shallow, do not linger to explore another. Like butterflies, from flower to flower, they pass on from one to the other, having known no one, but calling all friends.
Slowly, but subtly, I started to wrap myself up in myself. Layer, by layer, everything that was good in me, was only there for those who lingered long enough to peel back enough. And I thought that only fair, that only those who cared enough to know me, got me. There was precious little of me to go around and I kept it for those special people.
But as it turns out, *that* was a very stupid thing to do.
Sometime over the last year, reading the Bible at some hotel room somewhere in the country, I ran into something that clicked. Something that made sense and shone a light on the errors of my ways (uhh... no, I'm not going Born Again on you folks ... keep reading).
Mathew 5:15 Neither do men light a candle, and put it under a bushel, but on a candlestick; and it gives light to all that are in the house. Let your light so shine before men, that they may see your good works.
There's a certain arrogance of self that is required to be an introvert. That it seems worthwhile for someone who's met you once to dig through all your issues, fears and eccentricities, to know the really awesome person you are. That the end result, i.e You, is something enticing enough for a stranger to actually embark on that quest. As if there is some secret sauce, essence of pure self, that makes you unique among all others. That it doesn't really matter how you appear to be, that all that matters is how you really are. Pfft, maybe in an Apatow World.
Perhaps it is humbling to know that what you are isn't worth someone's time - at first glance. Perhaps there are so many who are boring on the inside & outside, that the odds are against you, all the way. Perhaps they are indeed shallow people who judge others by appearances or by popular opinion/reputation. I don't exactly know why people don't bother to look twice, but they don't (actually, some do ... which is how I got by for years). But it's not their problem that they don't, it's yours.
So, pull the covers off the true You. Shovel out a path through the icy reaches of your outer surface, put a window on your soul. It'll change your life.
In short, SUIT UP!
--Be yourself; everyone else is already taken.
-- Oscar Wilde
posted at: 01:01 | path: /philosophy | permalink |
You get to say no last goodbyes.
One cold december evening, I walked into a house of wailing to see my dad in a casket, laid out in the living room. If I could call it a living room anymore. And I couldn't even cry. Not while things were left to be done, people left to be comforted and a pair of large shoes to be filled, before I could truly mourn.
I cried into my pillow all night. I told myself that I wouldn't. But as the waves of sorrow came crashing down, I just couldn't help myself. I told myself that I had to be brave, I had to be strong, that now more than ever, I am a man.
The years have passed but, every december 2nd, I turn into that weak kneed young boy. Feels like it was yesterday. And I cry.
And I cry. For all that I've left unsaid. Respect, love and my last goodbye.
--For in that sleep of death what dreams may come, must give us pause:
there's the respect that makes calamity of so long life;
-- Shakespeare, "Hamlet"