Time Spent Programming

Life

Well, writing argumentative essays and doing math and being intellectual is fun and all, sometimes it's nice to just kick back and relax a bit, eh?

Working on Artificial Intelligence (actually "Pseudointelligence" allegedly; but AI sounds better), and having a blast not getting it done in code. These things get really complicated really fast when trying to transition between mathematical language and computer language; even more so when trying to get it into plain language, so I will not bother explaining it here.

Wasting time on purpose is a great way to relax. Clearing your mind out of all the loose little ends every now and then does wonders for stress. Have to steel yourself for senior year, eh? Between independent research, challenging AP exams, leadership positions, college applications, and social life, I might as well screw around while I can.

Speaking of screwing around, I might as well get something done for the Annual Roguelike Release Party. Expect to see something out of me September 19th, 2010. Hold me to this, please.

Meh, anyways, have a good last few weeks before Labor Day. I have to get some work done.

A Time and a Place

RescueTime Efficiency Summary:

-0.58 over 17h 8m vs. Avg. User 0.18 over 11h 10m

Sorry about that; bbl.

Writer's Block

What is writer's block?

Lack of skill, of concentration?

What lies between our conception and our paper?

... A writer's block?

Was there anything there to write in the first place?

How would I know?

Why can't I overcome this hurdle?

Am I good enough for it?

What do I have to do to see?

... Adequate enough?

When will my idea become clear?

The idea is mine?

Where is the inspiration, the light?

Is this the inspiration, the light?

Who is writer's block?

Me.

Web 2.0

Web 2.0; in an era where bandwidth is cheap, and storage space even cheaper, there has been a proliferation of blogs, news websites, and public forums where once there were none. Electronic development has given the many a voice that can be heard around the world; but at what price? When information is cheap, the emphasis is no longer on the information itself, but on those who present it. This is the problem with Web 2.0.

Web 2.0 has been, and will continue being, an amazing step forward in humanity's capacity to communicate. The fact that I can look up one of over 3 million articles on the Wikipedia and have the luxury of demanding that each point that I read be clearly cited with a respectable source is an advantage that I will not deny of Web 2.0. Or the fact that I can catch up on my friends with the click of a single button; no more rummaging through lost and broken contacts, scouring social networks for someone who might still be in contact.

Back in the "ye olde days" of the late 1990s, Yahoo had a single news portal, and that was all that was deemed necessary. Today, Google has a separate search algorithm for news stories, designed specifically to sort through the hundreds, if not thousands of mainstream blogs catering to news ranging from the intimately local to the expansively global. But, when attempting to do research for a political piece I was writing, it struck me how useless this search algorithm was.

On the sidebar of common options, Google News allows you to choose between three different categories: "All News", "Images", and "Blogs". Notice that a "News" category is missing? In an era when information is cheap, the emphasis is no longer on the information. Although I lack hard numbers to cite here, a Google News search will bring up around 9 editorials or other opinion pieces out of every 10 stories returned. When did editorials become so popular? My local newspaper runs around 2 or 3 pages of editorials out of a 20 page newspaper (not counting sports, lifestyle, advertisements, etc...). On the Internet, information is cheap; and if writers want to make a living peddling something cheap, they have to either dress it up as something more expensive, like an "opinion article", as if that added any intellectual value, or deliver it en masse; sometimes both.

Looking up information on the proposed Cordoba House to write an unbiased summary of its political effects on American politics, I came across the fact that the Anti-Defamation League had asked the Cordoba House to reconsider its location. This seemed out of the ordinary, so I searched this story up; lo and behold, scores of articles proclaiming the ADL's incredible hypocrisy, or how the ADL was ardently against the Cordoba House, or the reactions of the Jewish community on the ADL's decision. About two or three of these articles were polite enough to actually quote more than a line or two from the ADL's own words. About zero of them were polite enough to actually link to the ADL's press release, which I had to find myself.

Some pundits of the Internet claim the problem is that "we're drowning in information". To the contrary, I'd propose that we're drowning the information.

Back On Track, Part II

Let me introduce you to the concept of a transformation. In our limited definition, a transformation takes a set of coordinate spaces and maps it to a new set of coordinates. For example, a translation, something we should all be familiar with, takes a coordinate and 'slides' it across. In mathematical parlance, T_{(u, v)}(x, y) = (x', y'). But there's slightly more to this story. A transformation is a function, and adds a new set of equalities. In this case, x' = x + u and y' = y + v.

I said in the previous post that the point (X, Y) was the center of the turning circle, and that we would prove that here. We actually don't have to; these are defined to be the center of the turning circle by definition of a circle using parametric equations. However, to find this coordinate is slightly more difficult.

Consider in a real world situation, we would only have x(0), y(0), and θ(0), the starting conditions of the vehicle. But the equations we have derived demand (X, Y) and thus we have to solve for them. But there's actually a slight caveat that I did not mention earlier: notice the specific definition for s(t). We defined that at t = 0, the vehicle starts with a 'clean slate'. That the initial position is zero, the initial distance traveled is 0, and the initial angle is 0. This is not necessarily a bad thing, because we can finagle reality into bending to our wishes. (By the way, the value for (X, Y) is (0, r(t)), which is fairly trivial to calculate under our premises; just simply plug in appropriate values, i.e. t = 0, x(t) = 0, etc... to find these values).

Let's define four new variables, t_r, x_r, y_r, and θ_r, which represent the 'real' initial x and y coordinates and the starting angle, respectively. These might be CPU clock time, GPS coordinates and angle relative to the equator; or unix timestamp, coordinates relative to a nearby building and angle relative to said building; the actual values of these numbers can vary wildly. In order to deal with this, we will first define a transformation that maps t' = t - t_r, another transformation that maps x' = x - x_r and y' = y - y_r, and finally a transformation that maps θ' = θ - θ_r.

There is one special condition. We can translate time and spatial coordinates separately from each other because they are separate dimensions, i.e. for the purposes of translations, they act independently of one another. However, θ is a different story. When we do such a transformation, we aren't actually doing a translation: we are doing a rotation. Sparing you the details, just suffice to know that using a rotational matrix, you find new equations x" = x' cos(-θ_r) - y' sin(-θ_r) and y" = x' sin(-θ_r) + y' cos(-θ_r).

Now, there are no caveats to our equations. Everything works out beautifully, now that our equations are done in terms of t', x", y", and θ' instead of t_r, x_r, y_r, and θ_r. To convert between the numbers our formulas output and numbers in the 'real' world, simply apply the inverse of these transformations backwards. That is an exercise left up to the reader; have some fun trying to figure it out.

The important thing to remember is that you can always convert from one system of coordinates to another by replacing the variables by their original mapping. In our various calculations, we could have replaced every instance of t by it's actual value, t - t_r, but you can see how quickly that would get even more confusing. Mathematicians strive to put constraints on problems so that they are more manageable; it is the same reason that up to this point, we have been assuming speed and angular velocity are constant.

So, what if we stop assuming speed and angular velocity are constant? Well, I don't know. The farthest I've gotten in the formula is actually only one step away from our formal definitions in the previous part:

In these equations, I have removed the superfluous (t)'s from all of the functions, mostly just for aesthetic concerns, and I have reexpressed the integral in terms of θ. The chain of logic is actually pretty simple: rearranging dθ/dt = ω, you get dt = ω^-1dθ. Replacing dt in the original integral gets you the formulas above. But it isn't actually any closer to solving the problem. Only when you are given more constraints, such as the value of ω, or dv/dt, also known as the acceleration, can you begin to work somewhere else. Or not.

Some might paint me as dejected for not getting anywhere. But quite the contrary; I could not be happier. Mathematics is not always in the answer; sure, the rush of epiphany is unparalleled. But the true object of mathematics is to push one's own boundaries. I learned a lot attacking this problem, tracking down false leads, making mistakes, referencing material. But even though I have not succeeded, I have learned a lot, and I have pushed my boundaries. Isn't this all that we can do?

Back On Track, Part I

Sorry about that. Back to normal, scheduled updates.

Let me introduce you to one of my favorite subjects, math. I've been working on a problem interesting to me these past few days:

Say you have a vehicle traveling forward at a speed given by v(t), that can only change its direction by turning with an angular velocity ω(t). What is the equation describing the position of the object, , at any given time?

The reason this is so interesting a question is that it involves combining both linear and angular motion; a complex and non-trivial task. First, let us set some constraints on the problem to give us an idea of where we have to go: let us set a constant speed of V and constant angular velocity of T, to simplify our equations vastly. We can easily see that the total distance traveled is given by the following equation:

By similar math, the vehicle will have traveled a total arc of θ(t) = Tt. But that is not enough to fully describe the motions of the vehicle. One more piece is needed, the turning radius.

Consider for a moment the formula s=θr, the formula for the arc length of a circle. But what does this have to do with velocity? Assume with me for a moment that the vehicle is moving along a circle (a fact we will prove in a little bit). Using the formulas we have defined earlier, we recognize that s(t) = θ(t)r. We have left the radius in boldface just to illustrate how the parts fit together. Defining some equation r(t) as the turning radius, and solving for r, we find:

It is a reasonable to assume the vehicle is moving in a circle, but assumptions are not good enough for mathematics. We will show that the vehicle is actually traveling in a circle. More rigorously we define x(t) and y(t).

Why? The actual velocity of the vehicle at any time t is a vector with an angle of θ(t) and a magnitude of v(t). Notice an important distinction; speed is a scalar quantity, and hence there is no arrow over our speed function. Velocity is separate from speed. The same goes for the position  and the x-coordinate x(t). The appropriate x and y is obtained by integrating the respective components of the velocity vector, namely the formulas you see in the integrals. Once again, we will assume a constant speed of V and a constant angular velocity of T. Integrating these formulas out using previously discussed, we find that:

This is actually a tricky answer. The naive interpretation of "moving in a circle" would be to assume that the x component is found by multiplying the radius by the cosine, and the y component by the sine, but one of the harshest lessons mathematicians learn is to never place too much trust intuition. It can point you in the right direction, but it is no substitute for proof. I actually fell victim to this. There is one point of interest in these two equations, the two constants of integration. The point (X, Y) is the center of the turning circle. To discuss proof of this is another non-trivial problem which will lead us into tomorrow's discussion.

PS: Images are kind of mangled in this post. I had to tinker around with blogger for a little bit before fixing them. Next post will have functional images.

Identity

Are we meant to be individuals? Or are we to be some part in a greater existence? Should we remember where our roots were sown, or should we keep forward and never look back? Who are we, really, and how important are these questions?

Do successes become greater with context? The story of adversity overcome, of tenacity and grit in the face of all odds, is an appealing one. If the New York Times bestseller list is an indication of anything, at least. But where does identity lie - in the journey, or in the destination?

I've seen both ways of looking at identity. It is necessary to move forward without looking back to escape an identity we do not want to be associated with; living in a mixed-income neighborhood has shown me this. But, to never look back is to spurn those who have brought us where we are, losing important perspectives on our lives - something which can be more important than any achievement. A new perspective allows you to see things from a new light; which is sometimes all you need to see.

It is nice to think that there might be simple answers to these questions; but sometimes these questions prefer to stay unanswered. The only thing we can hope to do is look inwards and define these things for ourselves, defining ourselves in the process.

A More Sane Update Schedule

Updating every weekday is more insane than it seems. Please suggest to me a new update schedule. I'm thinking Monday-Tuesday-Thursday with special features occasionally on Friday.

More thought poking will continue tomorrow.

Norman Plays Grieg

This reminds me that I need to get hotlinking up. Here are two of my favorite pieces, played by yours truly.

Grieg Elegy Op. 38 No. 6
Grieg Waltz Op. 38 No. 7

Clarity

Words of five lack size
Ideas are best uncovered
Practice with few words.

Minds always wander
They travel any which way
Limits clear the path.

Guinea Pigs and Argument

Well, let me explain to you one thing: any post you see with the tag "essay" is written in 25-40 minutes; after this time period, I allow myself only minor edits. It is a self-imposed limitation, but a necessary one; I am a terribly slow writer, and fumble around with the English language often. One of the goals of this blog is to keep my essay writing sharp for future use. So congratulations, you are all guinea pigs. That being said, I encourage you to comment; not just to help me hone my writing, but also to serve another purpose.

"Nerds", in lieu of a better word, may at times seem overly harsh towards one another - especially those of us who have grown up with the polarizing influence of the internet, there seems to be no end to our arguments. This is neither a sign of harshness nor disrespect. We do not argue to be right, we argue for the sake of arguing, whether we notice it or not.

It has been said that it is only when a person takes a stand that their morals become evident; this is no less true for argument. When taking sides, the side we are defending becomes plainly evident to us. When we attempt to assert our opinions, we learn what our opinions truly are, exploring them in ways we have not before, and exposing them to perspectives we have not seen them from before, perspectives which we assimilate.

This is a very personal process, and is hard to relate with salient examples; but think of every time you have argued with a more experienced person, and discovered a flaw in your understanding - I'm looking at you, every single high school physics student ever. Or think of every time you have argued with a less experienced person, and had to strengthen and improve your knowledge in order to explain things clearly - try explaining the high school experience to a younger peer, and you find yourself consolidating your knowledge and truly noting what is important and what is not.

Experienced people understand this, and use it to their advantage; but the key is humility. To quote politics, "don't get mad, don't get even, get ahead". Do not argue to try and convince the other person of the rightness of your position - rarely, if ever, does this happen. If you attempt this in every argument you have, then you will find that nobody is better off for it. Argue to bring your point across, and see your opponent's point. Argue to argue, and if you keep your head down, you might just discover something unexpected.

The Illusion of Hard Work

Hard work is hard work. Let me expand this simple statement: hard work is not a function of how long you have spent doing work, it is not a function of well you have completed your tasks; it is a function of how hard you work. Hard work is an illusion; and to overcome this illusion is to understand the true value of work.

You may be led to believe that hard work can overcome any obstacle - it is a fundamental tenet of nearly every ethic, and seems like sound advice, if only we had the patience to follow it. But to follow this "wisdom" blindly is not wise at all. The key to success is not hard work, but smart work. The key is to cut as many corners as allowed, and effectively use your time - if you can finish 90% of the work in 70% of the time, why bother with 100% of the work in 100% of the time? We grow up in grade school striving for that 100%, but we are then thrust into difficult situations, and we find that we have never learned how to rush and prioritize.

Perfectionism has its place. But efficiency is far more important. Hard work is a means to an end; but it is not an end in itself. Spending hours studying in the library is not noble - far from it. The mark of a successful worker is not how long they need to work, but how long they choose to work. Be sure to always be aware of that difference.

(P.S. Yes I realize this essay sucked.)

Spark of Genius

I was considering a subtitle of "How to Reliably Screw Yourself", but I prefer a little bit of subtlety.

It's been said that it requires 10,000 hours of effort to become a master of something. Although modern life may give us a semblance of control, everything still boils down to cause and effect. We are not necessarily any more in control than mice in Skinner Boxes. Yet every now and then, we see through the code. We have an idea, or a breakthrough, that's bound to change ourselves and the world. We find the last little piece of the jigsaw, and our plans are set into motion. All that separates us from our goals is hard work, right?

To think that an idea or a breakthrough might just change the world implies that we have more control of the world around us than we actually do. The world is not built on ideas, it is built on hard work and progress. Take the building you are currently in - did someone at some point have the brilliant idea of designing the exact building you are sitting in? Or was it something more mundane?

There is a conception that there lies a barrier between the mundane and the enlightened. Those with the idea and those without the idea; that we are separated by nothing more than jigsaw pieces. Many times I've seen people who complain that they just do not get "it". That perhaps, if they had that one little piece, they would be better off. Entire businesses are built around this idea - that all you need is tax counseling and you'll become a better buyer, or if you purchase turf your lawn will become beautiful, or if you buy GenericCorp Product™ your life will suddenly become better. Unfortunately, the world does not work like that - and to think that it does is a surefire way to screw yourself over.

Ideas are like lighthouses. They shine brightly against dark backgrounds, making them clear to us. Yet to navigate by lighthouses is to miss the features of the land around it - that is, when you emphasize that one idea or that one breakthrough or that one item, you will always lose sight of the bigger picture.

When you're learning a difficult school subject, by concentrating on the concepts but not the problems, all you learn is meaningless drivel. Too often people forgo problem sets because they feel they need to only learn the concepts before they're home free - which might work for the kind of shallow learning we do in grade school. But to truly navigate the land, or to truly learn a subject, you must learn to navigate by the land, not by the lighthouses; it is an exercise left up to the reader to figure out what this means.

Genius does not rely on the spark of genius - quite the opposite, in fact. True genius relies on being able to trail-blaze; to be able to walk the land without any guidance, and move forward, with or without "the idea".

Announcing Project: A Rustle in the Leaves

Hello World!

I'm now unveiling a project I have had planned for quite some time now - a personal blog to serve as both a soapbox for my views and a snapshot into my life. While I get the rest of this site in order, I suppose it's time I introduced myself.

My real name is Norman Mingchen Cao (or in school, Norman Cao); or 曹明晨 (pinyin: cáo míng chén) in Chinese. I've gone by multiple aliases in the past, but generally speaking I like to call myself Normandy in these waters. Don't bother searching for me - my alias is drowned out by references to a certain city in France.

But what's in a name? How well do you know me? In this initial post, most of you reading this now probably happen to be my friends - but past my handle, how well do you really know me? Can you name my favorite color? Do I prefer rice or noodles? What do you know of my hopes and fears? How about my motives, and my aspirations? Truthfully speaking, I am probably one of the most opaque people you have ever met.

Join me then, on this journey. Who knows? Perhaps you'll see a rustle in the leaves, and uncover something new. After all, a lot more is in a name than you might think.