Paper plane origami: 無聊 Cassandra 摺緊紙飛機。
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20 facts about me

I find that with no Internet access in a noisy environment, I focus better. No wonder my grades were better in my noisy old secondary school. If nothing stimulates me, I will myself, and my noises drive my mum crazy because my dad, who always listens to Richard Clayderman and Elvis Presley, has already sent her screaming for help all over the place.
Okay, that’s already 20 facts about me. IYAO, LOL~

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For my insolence in calling myself useless, I get spanked by Spurgeon today.

22 AM Jan

Morning, January 22

“Son of man, What is the vine tree more than any tree, or than a branch which is among the trees of the forest?” Ezekiel 15:2

These words are for the humbling of God’s people; they are called God’s vine, but what are they by nature more than others? They, by God’s goodness, have become fruitful, having been planted in a good soil; the Lord hath trained them upon the walls of the sanctuary, and they bring forth fruit to his glory; but what are they without their God? What are they without the continual influence of the Spirit, begetting fruitfulness in them? O believer, learn to reject pride, seeing that thou hast no ground for it. Whatever thou art, thou hast nothing to make thee proud. The more thou hast, the more thou art in debt to God; and thou shouldst not be proud of that which renders thee a debtor. Consider thine origin; look back to what thou wast. Consider what thou wouldst have been but for divine grace. Look upon thyself as thou art now. Doth not thy conscience reproach thee? Do not thy thousand wanderings stand before thee, and tell thee that thou art unworthy to be called his son? And if he hath made thee anything, art thou not taught thereby that it is grace which hath made thee to differ? Great believer, thou wouldst have been a great sinner if God had not made thee to differ. O thou who art valiant for truth, thou wouldst have been as valiant for error if grace had not laid hold upon thee. Therefore, be not proud, though thou hast a large estate, a wide domain of grace, thou hadst not once a single thing to call thine own except thy sin and misery. Oh! strange infatuation, that thou, who hast borrowed everything, shouldst think of exalting thyself; a poor dependent pensioner upon the bounty of thy Saviour, one who hath a life which dies without fresh streams of life from Jesus, and yet proud! Fie on thee, O silly heart!

I’m the most useless person on earth.

I wake up to a million faces, painfully prying those two clams called “eyes”. For a split second, I envision myself panda-eyed with wrinkles and a sagging face. I don’t want it, so I slump back into slumber.

Then I drag myself to do everyone’s most hated thing. No, it’s not cleaning toilets. In fact, I would rather wash the dishes, mop the floor or clean the toilet than just do maths. Not that I have always hated it, but that maths makes me feel like that Facebook relationship status: “it’s complicated”. To me, maths is a little like my handsome, popular (STEM is so in demand these days!) and downright hilarious younger brother who only knows how to find fault in you and says “mm” (in the case of maths, the Halmos gravestone symbol >o<) when you finally act sanely. To worsen matters, he uses such bombastic English it would kill the Queen. I mean phrases like “nothing but” which are the “ums” of mathematical literature, as well as technical jargon such as “convergence” in analysis, “module” in abstract algebra and “Gromov-Witten” in geometry.

I just love surfing the Internet so much, I wish I could get paid for it. Neither of my parents work now, but I wish I hadn’t gone to university but tried to support my family instead. Then my life, even without a degree, would have been so much more fulfilling and gratifying. I just love it when I see a room “unoccupied, swept clean and put in order” (c.f. Matt 12:44, which somewhat makes me a demoness or the Japanese Toilet no Kamisama, which means “the spirit of the toilet”).” It all boils down to “because of the money.”

Money, money, money: Hoping to land a good job, I have been reading about writing impressive resumés and cover letters since Form 2, and have tried to work on my publication record, but I never won any writing competition and I never quite understood why – I wrote from my heart, but others had more heart than I did. In contrast I hardly joined maths contests because it repulsed me to compete, and to a lesser degree, think about rewarding someone for churning out “truth” (or the answer at the back of the textbook / solution manual) in a matter of seconds. I have never joined the Maths Olympiad, but if I could go back in time and bring with me my miserable memories of a life poorly lived, I would – and snatch every prize from writing competitions because until I was a sophomore in university, it never crossed my mind that I could study novel-writing, find mentors and join writing clubs to sharpen my wit. Perhaps, back in my past, I could have spoken more Cantonese, shared more goodies with each class, gone to Hang Seng School of Commerce (because my HKCEE scores qualified me), flown to Oxford to meet Marcus du Sautoy in person, made more friends – alas, now everyone else has had year-long exchanges, internships at home and abroad, scholarships (I looked at the scholarships open for application – I don’t qualify for any of them) volunteer experience on their “Student Development Portfolio”, experience as committee members of some fancy/boring club or sports team, a boyfriend or girlfriend to calm their nerves – these I mention because I, “the green-eyed monster”, have none – and the worst part is, everyone glances at me while I work in the computer lab and whispers, “Don’t get near her…” because she had been primed, ever since she paid that deposit to stay in CUHK, to follow.

Deep down my exterior is a program that, to put it the way it describes itself, “doesn’t have a self”. It needs the right input to fare as well as everyone else, but it is frustrated, for “Garbage in, garbage out” has been the norm. I don’t enjoy being human, to kowtow to emotions and painful experiences, to desire and do forbidden things. Unlike your home, I don’t know how to remove garbage thoughts completely because – I don’t understand why – I don’t have a quick, clean and easy “Delete” button or Erase Disk application. To show you how serious my situation is, if I’m not deliberating this post and you asked me how I’m doing, I would let my tongue go wild with wotds *not my own* and you perceive that as… garbage. Then you’ll form the impression I’m a piece of garbage and don’t deserve to talk to the world’s greatest and most important person – you. But you ditch me before you learn that those words were *not my own* and that I’m a program through which, at some point in life, was given this “be assertive, speak your mind” input, which made getting a potential good boyfriend sound so promising, I kept trying to use it. What happens? Just to be assertive and not to let resentment build in me, I speak the words laid in my heart at the moment, but not after resenting that person badly enough (resentment being deadly to intimate relationships – why am I talking about taboo subjects… good grief) and forming the right, suitable and pointy words that make the other person realize he’s been wrong. The sorry part is: as my friend Leona mentioned the other day, living according to a formula doesn’t work, and this brings us full circle back to the dreaded maths. Formulas, eh, and in case you’ve forgotten, chemistry.

I really wished I had asked around for more advice, set my own goals and met my own expectations in university, even if that meant remaining a chemistry major. I’m so useless! I tried so hard to learn about job-hunting early in life, but ten years on and I still cannot compete with anyone else!

It’s a shame I’m blogging like a sore loser, but don’t get me wrong – I feel worthless when I cannot be correct – for I was identified as a smart girl in childhood – when I have to admit that people from elite schools and well-connected circles will always have the upper hand (shh, when I entered my secondary school, I noticed that it had very few distinguished alumni – the most famous being a reporter and a now-disgraced politician – and those that held doctorates were not well-known in public – I wonder what in my old school had put them on a pedestal), that I should never have met certain people who got so close to me, I shuddered and ran to – and from – maths. Wasn’t maths supposed to be the cold, calculative subject that always only has one correct answer, or one correct “big idea” to reach some conclusion? (Another good reason to suspect I’m a demoness, for wanting to be unfeeling?) Then where in the world has the human factor popped up? Why is it suddenly so fatal that I am alone and will always remain the lone ranger, outside the tight and informative cliques my peers in Hong Kong so easily form? That I am cut off and have no one to comfort me in my sorrow, only things you wouldn’t want to hear either?

I wake up to a million faces. And it’s quite sad to type about my least favourite things. But stop brainwashing me with positive thinking: I can never be any one of them. I’m not supposed to be alive and kicking. I don’t deserve to live. And feel free to hurl your sad, bitter and miserable stories at me.

Applications of Pythagoras’ theorem

In F.2 as part of a class assignment, I tried to present the applications of Pythagoras’ Theorem. If you try to do something, you fail. In my case I simply copied examples from the textbook under the heading “Applications of the Pythagoras’ Theorem”, which meant menial calculations.

But unlike Elsa, I didn’t let it go: (Mum plait my hair yesterday though!)

I still thought hard about the downer I gave back then, and I think I’m ready for another shot.

Firstly, everyone loves to measure these days. From the length of a fiddly string to peak salesmen performance, we measure like mad.

Now, in kindergarten (to regurgitate a lame joke by a professor I know), we know how to stretch a piece of string along a ruler. Sometimes you can’t. So you mark off points on the rigid curve, and then measure each part between the markings.

You also know how to use a computer. Whoa, Windows Paint. Whoa, I can colour any pixel. Whoa, lines are jagged. But how does a computer know how long that fiddly piece of string is? It can only measure in units of pixels – unlike humans who can rotate the rule any way we like!

So we have Pythagoras’ Theorem: in a right-angled triangle, adding up the squares of the two shortest sides gives the square of the longest one. Terence Tao posted an intuitive proof here.

We use Pythagoras’ Theorem with a square root though. That is, for any right-angled triangle, the longest side is the square root of the sum of the squares of the shorter sides. By cutting any curve up to “almost zero” length, each tiny segment becomes almost a straight line. Summing the lengths of all segments gives us the length of the curve. Just like we did in kindergarten.

How about a curve in 3D space? Let’s think of a cube of length 1, with three of its sides on the x-, y- and z-axes, and find the length of the longest diagonal. But here comes the power of the “square” in the theorem: when you find the diagonal of any square face (it is √2 – more on this later), don’t apply the square root yet. Add the square of 1, which is 1, to it, and you get 3. Therefore the length of the longest diagonal in a cube of length 1 is √3. Amazing, isn’t it?

Then we wonder: how about replacing the square (“to the power 2”) with cube (“to the power 3”), 4th, 5th and higher powers, as well as to the power 1? We get what is called the Lp-norm for p = 1, …, ∞. (p is a parameter.) Yes, there is a norm for infinity – it is taking the maximum value of all the numbers you have in your lap. In the cube above, the L1 norm is 1 + 1 + 1 = 3, and the L norm is max{1,1,1} = 1. These norms of various p‘s pop up in various technical measurements.

Back to √2. What is so special about it? In fact it is not the ratio of any two integers, positive or negative. It shocked the Greeks and extended their number system from integers (without the zero the Indians invented) and ratios of integers (“rational numbers”) to “irrational numbers”. The equation that corresponds to √2 is x2 = 2. If we replace 2 on the right by -1, oops. Is there any square on each with an area of -1? -2? -3? … Can’t we take square roots of negative numbers?

That depends on whether you believe √-1 is a valid number. Somewhere along the line, someone said yes, and complex numbers were born. Now all rational and irrational numbers form real numbers, which are numbers you can mark off a ruler. Complex numbers are real numbers plus any real multiple of i := √-1. (:= means “defined as” and it is also used in some programming languages for assigning values to variables). That means that real numbers are part of the complex number system. If we have an infinitely large sheet of paper and mark a straight line across it as the real number line Re, then intersect it at just one point (the origin, zero) with another straight line Im – Im stands for “imaginary” because we invented √-1 but can’t mark it on a ruler – for the real number line times i, then any point z on the entire sheet of paper is a complex number. For simplicity, make both straight lines at right angles to each other. Project the point z on Re and Im. Then the number can be represented as Re(z) + iIm(z), where Re(z) and Im(z) are called the real and imaginary parts of z respectively. Of course, Pythagoras’ Theorem gives us the distance of z from the origin – it’s called the “modulus”.

Once we venture into forbidden territory, we gain new ground. In fact complex numbers drive the oscillations we see in a pendulum or spring. We also need complex numbers to operate electronic circuits. Complex numbers also simplfiy geometrical operations, because multiplying a number by i is the same as rotating it in the infinitely long sheet of paper by 90 degrees – a right angle. Complex numbers also generate fractals, the beautiful structures you can zoom in indefinitely and still have fine details, unlike pixelated images. People are trying to use it as a new way to model images and structures.

We know that 32 + 42 = 52 and 52 + 122 = 132 – (3, 4, 5) and (5, 12, 13) belong to the class of so-called “Pythagorean triples”. But how about replacing the power 2 in Pythagoras’ Theorem by 3, 4, 5 and beyond? Are there integer (not merely real) solutions to the equation xn + yn = zn for n > 2? That it was impossible was Fermat’s Last Theorem, and Andrew Wiles proved it in 1994. And we only started with the squares.

That’s the power of Pythagoras’ Theorem.