Grades

This is a post mostly directed at my parents, who probably don’t even read this blog. You could sort of call it a rant, I guess.

I have never been a particularly studious person, at least in relatively mundane subjects in school. Of course, I do my homework and stuff partly to learn the material as I go and of course to get the points, because grades obviously matter to some extent. You could guess that my grades are not all A’s at this very moment, or else there probably wouldn’t be anything to write about.

But there have been various accusations that have caused plenty of arguments around my house. I address them as follows:

1) YES, I know about my grades too. I get emails from Schoolloop every weekday. I usually don’t check the progress reports unless something major has changed. But when something major has changed, I will want to see what happened for the better or for the worse. And many precipitous drops can be explained by…

2) Ok, it’s a zero. Wait… I turned in this assignment! But of course, you’re going to call me a liar because the teacher knows more about me than I do myself. Please note that teachers have a fairly large latency period and I communicate to them about grades in the following manner:

a) The day after I see that the grade dropped, I’ll go contact them. Of course it’s patently ridiculous to expect instant gratification. But my parents expect it anyway.

b) I’ll then probably wait a day or two to see if the issue resolves. No, I won’t ask them every single day. Teachers in my opinion are people and are not machines that do things instantly. 

c) I ask people relatively non-intrusively. This is to add on to the unspoken politeness I guess. MAYBE it lowers the chance of it actually getting changed, but I think cordiality is an asset that is realized in better subjectivity.

3) Maybe the zero is actually a zero. Well, ok I admit that I forgot to do the assignment. I can’t remember everything. Of course you can tell me to organize better, but I’ve done that before. It doesn’t help that significantly and I believe it poses a significant burden. That is to say, yes, still zeros exist. Yes, I write assignments down if I feel that there’s a lot of things to do for the next day. 

That said, I believe the number of legitimate zeroes that I have had has sharply declined in the past few years.

4) It’s near the beginning of the school year! The teacher decided to change the point system into a category-based one. (or conceivably the other way around but I don’t think that has ever happened) This can wildly affect grades and clearly it’s something to comment on. NOT. I really don’t want to spend another half hour explaining what actually changed, so stop getting onto my nerves about this issue.

5) The grade has still been [insert not that good of a grade] for the past week! Yeah, as if it’s my fault that the teacher either doesn’t update much (Bellotti updates, what, once a month?), or there really wasn’t much to add. I really can’t affect the updates if there are no updates.

6) YES, I do care about my grades. Maybe not with the same enthusiasm as you, but I recognize that an A is better than a B. I will attempt to get an A without spending an overly large amount of effort doing so. 

7) While I appreciate you caring about my grades, and I guess your possible stress over the grade rollercoaster is possibly justified, please note that this stress is entirely optional. If you need to relieve yourself of this stress by ignoring the grades for awhile, DO SO! I actually encourage you to do so. I look at the grades every day, so I know about my situation already.  I think I can handle this “stress” relatively well, but it seems by your outbursts that you guys might not take stress nearly as well. So… yeah. You don’t have to be a proxy for me, especially if it leaves your health/emotional stability in the worse. 

kthxbai.

Research Complete 2 – A design philosophy

I really hope that this idea eventually gets implemented someday (also hopefully by me). This would be for a computer version as a forum really wouldn’t do the calculations justice. It’s sort of my idealized vision of what a civilization-building game would go like, and I’ve played quite a few civilization games (and even their modded counterparts) – Civilization 3, 4, 5, Rise of Nations, Age of Empires 2, 3, Europa Universalis 3, Victoria I and II, ostensibly Starcraft, Starcraft II (to a very limited extent), Warcraft III, and a bunch of mods for Civilization 4. I particularly enjoyed Civilization 4 and Rise of Nations. Of course, I like the Blizzard crafts as well, but they’re less related to a semihistorical context.

EDIT: I forgot to mention Sins of a Solar Empire (not relevant here) and Galciv 2 (very relevant to some of the ideas).

I would suggest at least getting a feel for Research Complete (1) which can be found in the AoPS forums. It was designed to have a heavy emphasis for research with a branching out as enabled by tech.

RESEARCH

Not much I would change here. It would probably have to be a tree-like structure with some techs unlocking other techs. Since I probably want an RTS feel, I would have to ensure that the time standards are reasonable. Early techs would take a few minutes to maybe half an hour, and later techs may take a few days. That would be if the game is persistent multiplayer, which is probably the most reasonable genre to host such a game. (that said, I’ve played quite a few persistent multiplayer games (tribalwars, ikariam, civworld) and they all seem to play the same way. I want to break out of this mold).

I like to keep a mystery of exploration. The first time RC was created, nobody knew what was the best thing to maximize, and this mystique keeps the game interesting. If things are consistent and never-changing, people quickly determine the best possible way to do things, the fastest way to do things, and do them. They make guides: “oh, do this and that and then that for the fastest possible victory” and the game becomes boring and solved. Randomness is one way to do so, but it has to be implemented in an interesting fashion. AI would be one (although the investment to make a worthwhile AI is probably too much to ask for). However, I believe that it may be worthwhile pursuing RANDOMIZED TECH TREES!

They would still make sense, and the gameplay would still have to be viable – core gameplay function should still be intact, although some secondary modules could probably be done away with as an interesting twist. A game without most military function would certainly change some warmongers’ strategies. Also, you would not be able to see most of the tech tree until you unlock them – this is an important characteristic of RC which largely prevented such tunneling.

However, people largely adopted strategies from other people because it was too tedious to copy-paste everyone individual data every day. So people would know what everyone was doing very quickly on. This can be mitigated with a computer version, a persistent multiplayer online game. Collaboration to explore the tech tree is definitely a possibility and I would encourage such alliances. But to prevent large alliances from fully traversing the tree the tree would have to be somewhat big.

(New!) TOOLS

This would play out similarly to Doodle God. It’s a simple minigame to pass the time, and this time there are actual effects. You can unlock more starting items with more techs, of course. Timesinks like these can reward active players. Of course, Doodle God is also susceptible to walkthrough-ization, but hopefully randomized tech trees mitigates this (so random items). This provides a crafting aspect to the game, and perhaps this can be an itemsink: you might damage a resource beyond use if you try to combine things incorrectly, and it might take time before you get good at combining even correct recipes (think of it as learning)

(Sort of new!) LAND

In RC1, this manifested itself as the construction module, but now I’ve split this up into two modules: the use of open land for raw resources (such as metals, rocks, trees) and expansive plots (like farms), and the urban center filled with lots of buildings. I neglected the land aspect in RC1; I hope to include it in RC2. Notably, you will need lots of resources to fund your little city, which comes from the use of land. You can apportion a certain number of people to farm, others to mine, and so on, and in the land module you do not make buildings but rather improve infrastructure. (maybe improve your mining capability or something). Land will have some sort of terrain modifier that makes certain lands more conducive to mining and others for farming or quarrying. With that said, food is important for growing your population which is important for the workforce.

CONSTRUCTION

And this is the building part of things. Buildings will involve the conversion of lower resources into bigger and better resources.  They require people to operate (of course (ok, maybe until you get robots to operate the buildings for you)). You have a lot more control here: you can set the number of people in each building which affects the efficiency of each building.

I like the construction space gating mechanism as much as the next person, but it makes less conceptual sense if we are operating on the premise of such expansive lands. While there will still be a land cost, it will be rather minor. The real cost comes from the construction costs, and the costs to house people into these buildings. Efficiency will operate on sort of a logistic curve: it takes a critical mass of people for a building to become effective, but stuffing too many people into a building and you run into diminishing returns (and perhaps people might die due to bad conditions, etc.). So one would need to create duplicates of the same building, i.e. expanding the building outwards. Food will be a more relevant concern.

MILITARY

I had a dream last night. It would make the game rather simple, really. And that is to create the concept of NEIGHBORS. What I didn’t really like was the monotony of the maps found in Tribal Wars and Ikariam. Each island looked the same, every village was so perfectly aligned to a grid. I get rid of the map, at least partially. Everybody, upon starting the game, is assigned a few (perhaps 6-10) neighbors. These neighbors are not all neighbors with each other though! Instead of a cluster system, there is an abstract map where you have neighbors, which is probably assigned partially based on your join time (so that your neighbors should be of approximately the same relative strength). You can only perform your military exploits on your neighbors (although you can resort to more peaceful measures such as trading and such). However, the catch is this: when you inevitably try to expand your land area via military exploits by carving out land from other people, you might acquire additional neighbors, who could possibly be quite large! The number of neighbors you have should roughly be proportional to the perimeter of your empire, which can either be well-managed or perhaps less well-managed.

Military actions are still much of the same: raid for resources, conquer for some lands, attack for actual military conflict, etcetera.

ESPIONAGE

This was a thorn in the foot in RC1. But with an online interface (and with an actual need for espionage with hidden values and such) espionage might actually work well here. With more powerful spies you can get more information from your neighbors.

MARKET

Another painful aspect of RC1. You can offer resource trading between neighbors and probably a range around them (so perhaps a 4-5 neighbor range). It won’t be a stock market implementation (until perhaps later with stock markets). It could also function in that you have a certain resource that someone else doesn’t have, and a trade could occur.

INVENTIONS

Now, inventions are tied to technologies, so you won’t get irrelevant inventions or anachronistic inventions. You can only invent things that you have already researched. I think Victoria II does a good job exemplifying this aspect.

OTHER MODULES

May add later.

English Homework

PRETEXT: There’s English HW and I decided it might be a good idea to post something after a ton of time of not posting. This is not the longpost, by the way. This is probably very cynical, sort of irrelevant to the prompt, perhaps grammatically incorrect, and I did no planning or editing (other than this PRETEXT) although I did backspace a few times to actually make sentences or catch spelling mistakes (which I habitually do, although this was discouraged in the context of the assignment). Hence, no closure. And stream of consciousness.

 

Are you prepared for the next step? (20 min)

In this essay-thingy, I will attempt to answer a question in 20 minutes. This is for English homework, by the way. The question is as stated, “Are you prepared for the next step?” No planning or editing whatsoever (although my pride sort of cheats this by backspacing quickly) But I digress. One minute in. The real world seems like a whole new world. There is no obvious shelter that is provided, and you need to provide it yourself with your own hands and labor. It is this that this is very different from the circumstances that I face today. Of course, there is college to bridge the gap.

Bridging the gap is a very overused expression. In 3rd grade, I’m pretty sure they said that 4th, 5th, and 6th would be used to bridge the gap to junior high, and in 6th grade they said that junior high was the bridge to high school. In a sense, yes. In a sense, no. I did not really feel the accomplishment of actually crossing the bridge – it was so streamlined. It feels rather as if instead of crossing a discrete gap, a river to cross, I was seamlessly moving from one to the next. I’m pretty sure the bridge is supposed to have this effect. But I never actually feel as if I actually crossed this bridge.

School has been much the same for all these years. While the addition of the period system introduced a variety of teachers that I hold a varied amount of respect to, it really wasn’t so different from before. After all, in elementary school we had to go to different “labs” every week or so for “specialized activities”, and inclass we definitely did a whole host of things each day. It’s basically a schedule. Scheduling appears to be a very integral part of college and beyond as well.

The commitments were basically the same as well. There are school assignments to do each day, and there are extracurriculars. However, testing largely phases out in college, with just a final (which of course has been introduced in junior high, since I don’t remember having to test for finals in elementary school). The claim is that grades now count for high school instead of junior high, which is in a sense true (after all it dictates which college you go to to some extent) but these grades are essentially thrown out in college, and life starts anew. Then college grades are thrown out after you get your second job. These grades are really quite transient, and nine minutes.

It feels as if the world doesn’t really need any given person. And in a sense, this is true as well. Your achievements are essentially for naught after a while, and as the ages pass, you die. Those lucky enough to be an impact to society pass on their legacy. But this is such a chancey thing. How will I know that my legacy will carry on hundreds of years from now? Science is probably the best way to get “immortalized”, and even we don’t know how immortal this is, as science (at least renaissance and beyond) is such a new concept, it’s only about 500 years old. But other things of relative time have mostly faded away. I wouldn’t be able to name most of the Renaissance leaders of the time, for instance. But in their time, I’m sure they felt a lot more important than the scientists did.

So there is the question of how to get a good legacy. Perhaps science will go out of vogue in another few centuries, but who am I to judge? I’m sure people will forget of basically everything in the event of a post-apocalyptic society. They will remember those who brought back any semblance of civilization, and clearly I should try to initiate an apocalypse and save a small collective of people (I’M JUST KIDDING). But really, does it matter what we do? Of course, our hivemind would completely collapse if everyone had this mindset.

If nobody thought they had any commitments, then modern society would probably collapse within a few weeks, if not days. It’s really quite scary how we, who aren’t hardwired to be excessively social (of course, more social than lone warriors like tigers, but I’m talking social at the level of ants or bees) manage to live in such a fragile weave of society. We do have primal, hardwired tendencies – greed, for instance, but society has managed to utilize these tendencies to incentivize stability to a sense. Money is definitely something to go after – the embodiment of wealth. But nothing in our minds has the inherent desire to become intellectual, and it appears as if this value is declining with the rise of the corporate world.

War – it comes up, and this is mainly due to our primal warlike tendencies. We were not meant to live harmoniously as a society of seven billion. But we somehow managed to do so, albeit quite shakily. And this is why war happens.

But back to the question, as I have like two minutes left. Am I prepared to enter this extremely unstable world? Yes and no. Of course, the phrasing “unstable world” should already sound warning bells, and of course I am probably not ready to handle such a thing. But in the end, does it even matter?

There are a few things that would indeed transform this mentality. Immortality would be something that changes the legacy problem on its head, and even the small changes that you make over time would add up to a large amount when you multiply this by an infinite value. But it might eradicate desire, as people lose motivation. But time is up.

 

A Quantum Leap

What an oxymoron. You know, how “quantum leap” in colloquial speech means “lots of progress” but quantum actually means small. Heh.

For those of you still eagerly awaiting for the post with a word count perhaps exceeding the total verbiage of the rest of my blog combined, this is unfortunately not that post. Notably, it is a more quantum post, both in size and by content. The longpost does not have significant quantum mechanics material.

Also, post restriction broken.

Disclaimer: I don’t claim to actually have formal knowledge of quantum mechanics. I won’t even claim to have any knowledge of quantum mechanics! So please be aware of any ramifications in using this potentially inaccurate information in serious applications, such as studying for a test, writing a research paper, or other life-changing events. Notably, much of this stuff was derived from a combination of websites and personal intuition.

Disclaimer 2: A lot of this stuff is review if you’ve already taken some sort of chemistry class. I’m sorry for giving you potentially inaccurate or redundant information. In addition, a lot of this stuff may be more focused on competitive-math-oriented subjects, so I’m going to repeat that information too. So that might be redundant as well.

It was a cool and rainy (if my memory holds on the weather that day) afternoon, about one year ago. In Honors Chemistry, we had been learning about orbitals. Now, the most important part about electron orbitals is not the naming. In fact, that probably holds for 90% of everything else, in that naming is generally not terribly important.

However, naming can be interesting, despite its lack of importance. In this case, orbital naming. No, not the s-orbitalp-orbitald-orbitalf-orbital naming (or else this blog post would be: sharp, principal, diffuse, fundamental. Ok this blogpost is done.), but the naming of each individual orbital.

We start with the s-orbital. I’m actually unsure what it’s named. Probably the 1-orbital. Since the names tend to be subscripts below the s/p/d/f designation, I think the only shape – that is, a sphere – would merely be an s orbital, sans subscript. Well, that was boring.

We could enlarge our orbital shell. The nth orbital shell can hold n supertypes (supertype being each individual letter/block) of orbitals. Yes, I’m being really nonrigorous with my naming. And that’s because naming is not important.

We call the s-orbital in shell number #2 the s2 orbital, the s-orbital in shell number #7 the s7 orbital, and so on. It turns out these orbitals are just larger spheres, with a bunch of inner spheres inside them. In a similar vein other larger orbitals are merely simple expansions, so I won’t delve too deeply into that subject.

The next one, the p-orbitals, is pretty straightforward as well. There are three types, the x, y and z. They are shaped like two-lobed thingies. I will now proceed to add a picture to this blog. This is unprecedented. Wow!

See? I was right about the s-orbital.

In other words, the p_x points in the x direction, the p_y points in the y direction, and the p_z points in the z direction. So far, pretty intuitive.

It’s interesting too, as all the p orbitals fit into a nice little compartament, along with the (not shown) s2 orbital.

Next, with some theory-crafting, although I’m guessing it’s well-explained already. Electrons, as you probably know, have several quantum numbers:

The principal quantum number, n, dictates the shell number.

The orbital quantum number, l, dictates what type of orbital it is. You know, the s, p, d, f, g, h, i, j, k, l, m, n, o, etc. thing.

The magnetic quantum number, m_l, dictates what shape of orbital it is. For instance, p -> p_x, p_z, p_y.

The spin quantum number, m_s, dictates which spin the electron is. As electrons are fermions, they have half-integer spins, i.e. either +1/2 or -1/2.

They are constrained by the following:

n≥l≥|m_l|

|m_s|=1/2

Notably, this gives us 2l-1 total orbital types per orbital. Moreover, this also produces the fact that there are 2n^2 electrons maximum in each electron shell. This also furthermore explains why the periodic table has 2 elements, 8, 8, 18, 18, 32, 32, … elements per row, in conjunction with the Aufbau principle. I always thought it was “Aufbau’s principle” but “Aufbau” means “filling-up” in German and therefore cannot really be in possession of a principle.

When l=0, we get the p_z orbital. When l=1, we get the p_x orbital. Or the p_y orbital. I’m not sure. The other one would be l=-1.

At this point, you ask yourself: why is p_z in the middle? Let’s keep going on.
D ORBITALS!

Come on, it’s not that complicated… yet.

We have five orbitals: d_xy, d_xz, d_yz all look like they have four lobes.

But what’s this? There’s ANOTHER d-orbital with four lobes? Yeah, d_(x^2-y^2). Are you even allowed to put such an expression in the subscript? Oh well.

Then there’s this fifth d-orbital that doesn’t have four lobes. Instead it has three balls stacked on top of each other, d_(z^2). Yay complications.

First, can you see any patterns? Unless you’re ridiculously good at inference, probably not. It took me up to the f-orbitals to figure out some meaningful trend. Why is this? Technically, I cheated here. The groups I picked are rather misleadingly grouped. As hinted in the earlier section, something is special with the z-direction. It becomes even more painfully obvious with the d_(z^2) somewhat sticking out. A few trends that you may notice, on the other hand: all subscripts are of degree l, so they’re all quadratic terms here.

Let’s reorganize the d-orbitals by the z-exponent.

5 d-orbitals and how they fit

I guess it might be somewhat a stretch to say that these clouds easily fit, along with s and p orbitals it would share. Although, I suppose the picture itself definitely does not fit the margins set by my blog. Darn.

So I’ll give instructions, by request, on how to view these oversized images. Open a new tab or window. Right-click the picture and copy the URL, and then paste it onto the tab. Your browser may have more functionality like “open image in new tab” when you right-click, so there may be more direct alternatives.

 

Yeah, d_(z^2) looks kinda like a normal p_z orbital with a donut in the middle. How confusing. Also, d_xy isn’t aligned to axes unlike p_x or p_y (while d_(x^2-y^2) is!) What gives?

At this point, the “Modern Chemistry” book stops here, leaving most students content and happy that they don’t have to learn more in chemistry. But this is rather unsatisfying. Exactly what causes this sort of behavior? What would the f-orbitals look like, and for that matter, the g-orbitals and beyond? How are they named, because they sure don’t look too consistent to me.

tada

Fun news! For the f-orbitals, two of them are named f_(x^3-3xy^2) and f_(y^3-3yx^2)! This obviously suggests some sort of pairing mechanism, and indeed this is the basis of the m_l being positive or negative. The other orbitals are named:

f_(zx^2-zy^2), f_(xyz), f(xz^2), f(yz^2), and f(z^3). The homogeneity condition still holds. The z-exponent thing also holds. (2 per z-exponent except when there’s only z^l.) Let’s try dividing out all the z’s and see what we get:

s: “1”

p: “1”, x, y

d: 1, x, y, xy, x^2-y^2

f: 1,x,y,xy,x^2-y^2,x^3-3xy^2, y^3-3yx^2

The pattern is pretty clear now, no?

Meanwhile, I’ll also try to describe the f-orbitals:

The two new f-orbitals have six lobes each. The ones with z^1 exponent have 8 lobes each, the ones with z^2 exponent have 6 lobes each, and the ones (or rather, I should say, one) with z^3 exponent has 4 lobes, all stacked on top of each other.

Since tabling seems to be a useful idea, we perform this again:

s: 1

p: 2 / 2

d: 3 / 4 / 4

f: 4 / 6 / 8 / 6

We know that those z^n orbitals all seem to have balls stacked on top of each other; it seems rather logical that the next one will have n+1 lobes, preserving the pattern.

But if we investigate the xz^(n-1) orbitals, for instance, we see that ALL of these double-lobes are stacked together. We can see the same thing start to form in the xyz^(n-2) and (x^2-y^2)z^(n-2) orbitals as well, with quad-lobes stacked instead.

And all of this stacking is iterated every time we multiply by z. With induction, we can probably name every single orbital…

Except for the orbitals with no z exponent to begin with.

Let’s take a look:

badly ms-painted work

They all have one layer, unsurprisingly (given that their z-exponent is zero). Starting from the p, d, f, orbitals, we see that the number of lobes increases by 2. They are equally spaced; anyone with decent math contest exposure should probably think about roots of unity at this point. [Root of unity: any complex number x satisfying the equation x^n=1 is known as a nth root of unity]  In fact, out of curiosity, let’s factor the d and f orbitals:

x^2-y^2 = (x-y)(x+y)

y(3x^2-y^2)=y*(rt(3)x-y)(rt(3)x+y)

When are these equations zero? For the former, (x-y)(x+y)=0 => x=y or x=-y. Look at the gaps between the lobes. Sure enough, these gaps exactly correspond with the zeroes! So the lobes can be seen as when the equation of the orbital is not zero. In fact, if we use the definition that these electron clouds are merely probability distributions (it could be possible that an errant electron is somewhere else; it’s just very unlikely), then we can see that the regions where the equation has a high magnitude exactly correspond with the regions where the electrons are “likely” to be found. I would also guess that this is how two electrons can share the same orbital, with such parity.

This makes naming future orbitals rather simple. We’re looking at every other root of unity of the 2(m_l) roots of unity. You may have noticed, especially with the f-orbitals, that the coefficients somewhat follow the binomial theorem. And this is no coincidence; if you are somewhat familiar with math contests, root of unity filters can easily kill certain problems. For example, this one:

What is nC0+nC2+nC4+…+nCn? (n is even)

If n were odd, that would be strange as 0,2,4… tends to not have any odd numbers in this sequence. It would also be easier, assuming that I typoed nCn as “nC(n-1)”, where you could easily invoke a symmetry argument.

However, there is another way.

Recall that the Binomial Theorem states that (x+y)^n = nC0 * x^n + nC1 * x^(n-1)y + … nCn * y^n. By plugging in (1+1)^n, we get the well-known fact that nC0+nC1+nC2+…+nCn = 2^n.

However, how would we extract out every other term? We can experiment and try (1-1)^n. Ok, that’s pretty obviously 0, but what exactly is it by the Binomial Theorem? It is equal to nC0-nC1+nC2-…+nCn. Still not quite there yet, but if we add the two up we get

2(nC0+nC2+…+nCn) = 2^n so our desired result is 2^(n-1).

It is a more complicated path to compute nC0+nC3+nC6+…+nCn (n is divisible by 3). Sure, you could do so combinatorially (really!) but it’s not much fun. Roots-of-unity filters are better in this case. If you don’t do math contests regularly, try this exercise out first. It’s admittedly quite hard if you haven’t seen this before.

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[SPOILER]

Darn this isn’t a forum.

Ok, so let the two roots of x^2+x+1=0 be w0 and w1. (They should be omega’s but oh well.) One property that you might notice is that w0^2=w1 and viceversa (w1^2=w0). You might think that property, that two numbers are squares of each other, is rather weird (or you may point out 0 and 1 being squares of themselves, but that’s slightly different), but it’s easily seen:

w0=w1^2=w0^4 => w0^4-w0 = w0*(w0-1)(w0^2+w0+1). So if w0^2+w0+1=0, it satisfies that property. Yes, w0 and w1 are complex numbers, easily seen with the discriminant of x^2+x+1 being negative.

The insight: Consider (1+w0)^n and (1+w1)^n, along with (1+1)^n. By the Binomial Theorem, only the terms divisible by 3 align with each other (giving you 1+1+1), while everything else cancels out, as x^2+x+1=0.

You get something like (2^n+(1+w0)^n+(1+w1)^n)/3 as your final answer. In fact, the fact that n is divisible by 3 was not even necessary for this problem. You could clean up 1+w0 and 1+w1 in this case (they turn out to become sixth roots of unity) but I’ll leave the algebra for later.

[/SPOILER]

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That was somewhat of a tangent, sorry.

At any rate, what do the formulas remind you of? We have x^2-y^2, and xy. We have x^3-3xy^2 and 3x^2y-y^3. (If you think I’m scaling up and down arbitrarily, you’d be right, because as it turns out constant multiples don’t really affect anything.) Notably, they appear to “look” like expansions of (x+y)^n. However, there’s a catch. Both (x+y)^n and (x-y)^n have positive coefficients of x^2 and y^2 for instance. We wouldn’t expect x^3 and 3xy^2 to alternate signs either.

Insight: Expand out (x+iy)^n. The real part spits out one part of the orbital, and the imaginary part spits out the other. I guess motivation for this could be wanting to get opposite signs for x^2 and y^2 😛

However, it doesn’t answer a few important questions:

Question 1: How does this factor out into alternating roots of unity?

First, if you didn’t know, the n roots of unity have the form cis(2pi*k/n), because x^n=1 =>(r*cis(th))^n=r^n*cis(th*n)=cis(0)

cis(th*n)=cis(0) only when th*n=2k*pi => th = 2k*pi/n.

[cis notation: cos theta + i sin theta, i.e. polarizing rectangular complex numbers. Polar coordinates are easy to multiply as it’s just multiplying magnitudes and adding angles. Thus, [r*cis(t1)]*[s*cis(t2)]=rs*cis(t1+t2).]

So, this is equivalent to either Re((x+iy)^n)=0 or Im((x+iy)^n)=0. It may be more useful to convert x+iy into r*cis notation here (r^n doesn’t really affect things to be honest).

Re(r^n*cis(n*theta))=0 => cos(n*theta)=0 => n*theta = pi/2, 3pi/2, 5pi/2, …

In other words, theta should equal to (2k-1)*pi/2n, i.e. the odd 2nth roots of unity. Similar logic for Im, just use sin instead. Therefore, (x+iy) must have an angle measure equal to the 2nth roots of unity, so it must be a scalar multiple of a root of unity. Hence the gaps are where they are supposed to be. Yay.

Question 2: But I want a nicer formula!

Well, that’s not a question.

But sure. Do note that the polynomial looks like normal binomial expansion, except with every other term…

Wait this sounds like that root of unity filter I talked about earlier! It might not be so tangential of a tangent after all!

But wait. It’s alternating signs! Let’s root of unity filter again, with different filters.

(1+i)^n => 1 i -1 -i 1…

(1-i)^n => 1 -i -1 i 1…

This seems to work; the i’s cancel out, and every other term alternate signs as we want.

So, the proper polynomial would be ((x+iy)^n+(x-iy)^n)/2.

———————————

Now, this explanation is only satisfactory to a certain extent. For instance, it still doesn’t explain why the d_(z^2) orbital has that strange donut in the middle. Well, now is the time.

Actually, pop quiz because I feel like giving a pop quiz. And “since I’m such a nice person”* (*cue Bellotti) this pop quiz is entirely optional!

Here are some g-orbitals. Please name them, and give the magnetic quantum numbers of each. Have a nice day.

A

B

C

D

(Note: the weird gray sphere is the atom itself.)

[Answers:

a: z^4, m_l=0

b: z(x^3-3xy^2) OR z(y^3-3yx^2), m_l=+/-3

c:  z^2(x^2-y^2) OR xyz^2, m_l = +/-2

d: x^4-6x^2y^2+y^4 OR xy(x^2-y^2), m_l = +/-4]

It was a fine day, knowing what was the approximate shape of the orbitals and knowing every single name of every orbital! But then I stumbled upon this site http://www.uky.edu/~holler/html/equations.html and it was once again a state of agony. (As it turns out, the functions on the right are the wavefunctions themselves. I suppose armed with that information one could accurately construct models for all of the orbitals. However, the associated text to these equations, http://archive.org/stream/introductiontoqu031712mbp#page/n141/mode/2up, is supposedly an “Introduction to Quantum Mechanics”. It was also written in 1935, perhaps a decade after the leading scientists formulated quantum mechanics itself :P)

I have no idea how that’s an introduction. Either they had really smart people back then where even the average layperson knew triple integrals, or I guess it’s not really an introduction. But I digress.

Let’s try to tackle the donut problem. Why would there be a donut there? My best guess is that it’s sort of like a standing wave. You have positive and negative regions, but they just swap with each other, over and over again.

I’m out of witty captions today.

More f-orbitals

The gold regions represent positive and the red regions represent negative. Or maybe it’s the other way around. Either way, they alternate ad nauseum.

Also, there appears to be a sort of rotational effect as well, centered on the z-axis (of course, what other axis is so special?). Meh time to learn about sigma pi delta phi bonding.

I guess I’ll end this post with an anticlimactic link: http://www.falstad.com/qmatom/

Enjoy playing with that app.

Restriction

of the post variety.

 

It’s a subtle one, but it exists. Mostly occured from a rather “coincidental” observation from somebody when this blog was 3 posts long. So I decided to “retcon” (probably misusing the word here) my posting schedule to fit this one. Some of you already know about this, but to those of you who don’t, I’ll just say that a) it leads to somewhat long posting breaks and b) I can’t just publish whenever I feel like it.

 

Also, while I do queue up posts, my queue of completed posts is stunningly small (read: queue size = 0). I do have half-complete posts though. As of last count, four.

 

This post will be fairly short, because I need to finish typing this post in around 3 minutes. Now 2 minutes. Also, I guess the previous post was unjustifiably long, and way too personal and nobody should care about that stuff. Although I suppose this post is as well. (although not “unjustifiably long”. Unjustifiably short?)

Endurance

I don’t have any of it. Either physical or mental, as I have basically concluded.

So today was the Berkeley Math Tournament. Here’s approximately what happened:

BART @ 7:30: Load BART ticket, look semi-awkward waiting alone for rest of team. Overhear some person arguing with the station employees (what is there to argue over anyway?), meet up with some of the team. Surprise! Some of the team members that we regrettably either “kicked out” or volunteered away due to space issues defected to other teams. Funnily enough, one of these members (Kr.Bh.) was kicked out on the grounds of lack of school loyalty during a previous math tournament, SMT. Oops. So anyways, meet up with them, board bus (fine, train) initiate cardplaying. Teach someone how to play Napoleon (and you should too here: LINK. Warning: seizure alert, sorry if your eyes get blinded. Oh, and it’s pretty long. You might want to skip over the rules for Prime Minister (a Presidents variation) but that’s a pretty fun card game (at least to our mathematical clique) so yeah. We’re (well I can’t actually speak for our group but this is my opinion) open to people who want to play cards with us… as long as you somewhat know what you’re doing. Different people seem to have different thresholds of this “knowing what you’re doing” quantity though.) We later on transition to BS poker because it’s easier to play on a train, with not as much card-throwing. No, a high card does not beat a pair.

Berkeley @ 8:40: Walk out BART station. Al.Xu. insists that we go one way, but RH.Wu. and the rest of us go to other way. Apparently Al.Xu.’s way involves an escalator, and ours involves stairs. Exercise! After much undue “paranoia” about cars running over us (so it was a very good position to stay in the center, lest some erratic car comes hurtling through the directed roads in the wrong direction, although I do suppose that if that happened we were all screwed no matter what our relative position, especially if we’re on the sidewalk) we reach the auditorium. Hmm. Life sciences for a math tournament. Ah well. Play more Napoleon (teaching He.Ma.) and thus Napoleon is gaining ground in the math community. Yay. (We managed to make Prime Minister the SFBA exclusive card game the year before, and Idiot the year before that. (Mao and Uno precede this, although neither are very SFBA-exclusive or math-exclusive. Although I suppose you could make the same argument for Napoleon.) Both instructions in the link above.) Late double breakfast: acquire orange juice (but importantly, the plastic cup!) Plastic cups are not biodegradable, and thus do not taste very good. I do not manage to actually eat it, just rip it up into shreds. Oh well. Team registration occurs, and the Berkeley people are apparently more receptive on esoteric team name selection than the Stanford people. (We sent “·” (U+00B7, Alt+183) for Stanford, and “☺” (U+263A, Alt+1?) for Berkeley. The Stanford people, to put it simply, were not very amused, and casted the name to “.”. Darn.) So anyways, we get t-shirts, nametags, the general stuff. Back to cardplaying.

Power @ 10:20: Yay combo power. Darn why did I pick extremely bashy stuff. Induction lemma, induction proof that uses induction lemma. Too much work for only 7 out of 90 points! Then write up 4 pages rigorizing something for 8 points. What time’s up already? Ok let’s just say that it’s equivalent to what we want to show, and get it over with. Not the best performance, but considering we basically dropped about 30 points already of unsolved problems (No, Je.Wu., #12 was not solved), 15/60 is pretty good, considering how freaking long each problem took. My hand hurts 😦

Expected contribution: 15 max, perhaps -2 for that other problem.

Team @ 11:30: Tried #10, give up too hard. Tried #9, give up too hard. Doesn’t look good so far. Ohey #8 is trivial. #7 is trivial. #6 is trivial. #2 is trivial and why did our team not get this earlier. #5 is a pretty quick mem, for a confirm. Oh, and relook at number #9, gee the calculation looks kinda bad, but YES WE GET IT IN THE LAST FEW SECONDS!

Oh huh so apparently I misread #6 😦 Consolation points for solving their intended question? Darn. That diagram was totally not symmetric.

Expected contribution: …36 what? (half a question for #5, #9 I suppose – calculation was partially outsourced to Pa.Ze.) This is out of the 8 correct = 72 points. (we did not get #10, and 42 was unfortunately not the right answer – though it was actually a semi-legitimate guess!)

Too much carry. Two good rounds in a row is a recipe for disaster. I haven’t been able to consistently operate throughout a full-day math tournament (including SMT in the year which I topped both subjects AT/geo; I basically slacked on team/power that year) so this looks like a bad sign. Also, I necessarily have to take significant ~30 minute breaks on the USAMO and other similar olympiad tests, or else really bad stuff happens, like when I basically space out during MOP test #2, getting 8/28. Oops.

(That 30-minute break thing doesn’t really work for these short sub-hour rounds.)

Indivs @ 12:20: Blast through first six questions. #7 takes some time, #8 takes even more time (1/3 of the test?). Then #10 was a pretty quick observation, and #9 I got a few cases but not very much else.

In theory.

As it turns out, I miss one problem for thinking that 47 is divisible by 3, one problem for disregarding the fact that 34 and 70 are not relatively prime, problem #9 for not seeing that a particular case was trivially winnable, and one problem for misinterpreting the wording of the problem majorly. gg

Ouch, that hurt. 6 => 12 contribution points there. 😦

———————-

So, that’s around 13+36+12=61 points out of a maximum of 300, so yep I’ve basically done my share and overdone by quite a bit. But at a cost of a pretty terrible individual round. Which is interesting in a team perspective:

Getting one team question correct is basically worth about four and a half questions correct. This is what completely roflstomped the SFBA A1 ARML team last year: the team round. (Their proportion is actually higher, rated at one team question per 5 individual questions) We got like 6/10. And that is bad – compared to most teams who got 8/10 or even better, that would be akin to throwing away one individual perfect score :O. Mediocre power round doesn’t help either, but I suppose there wasn’t too much improvement to be made there (Oh, and apparently there were some grading issues that caused us to get no points on #3? I don’t know). But anyways, the team round really really failed for us. I don’t think coordination was the big problem actually; we did satisfactorily well on the power round. My theory? We just get tired. Ok, maybe not “we” but at least I feel pretty worn out after going through the power round.  And thus don’t rely on me to get 8, 9, and 10 all at once, because that’s not how I roll usually.

———————-

Lunch @ 1:30: But no, that’s not the end of the tournament. Afterwards there was lunch. After another slight “snafu” where we only get  four bags of burgers, instead of five… and two of the bags don’t even have fries in them (Don’t get me started on the lack of ketchup on any of them), we finish them. The burgers were pretty good – they’re probably more worth it than McDonald’s burgers, although you still cannot beat Burger King onion rings, hah. (I have a feeling that my taste of taste is weird) For price comparisons, that cost us $6, fries (at least it should have) and drinks included. Eventually we get two more boxes of fries, and all is good. I’m actually unsure of how much a McDonald’s meal costs, but I’d hazard a guess of around $6 as well if you’re not going ultra-cheap. Seriously, these burgers were actually pretty decent. Especially the bread.

But anyways some people felt that the $6 cost wasn’t worth it, so they got their meal via other means, such as going out to downtown Berkeley and buying food elsewhere. Somebody got a sandwich, and that comes with forks and knives. Also, I think Berkeley is generally considered a greenish community, so of course these forks and knives were special – they’re BIODEGRADABLE! And biodegradable clearly equates to edible right? By the way, sanitation is not a problem as that knife was not previously used.

So I got a knife. Don’t worry, I was semi-bored and not actually hungry. At first I try using canines to etch out of the flat part of the knife, but that turned out fruitless (there was actually no fruit as well) and I only managed to catch a few biodegradable slivers. So clearly there had to be another attack point. Well, the serrations of the knife turned out to be small enough to be bitten off. Then start at the top of the knife and work down. Chew long and hard, and miraculously your saliva actually begins to dissolve the knife. The human body is amazing.

By the way, if you were wondering what it actually tasted like (because I assume a normal reader probably does not intend to replicate such a procedure), it tasted kinda planty. Not really like wood though, but it really did feel better than the plastic toothbrush I tried to digest about two years ago (which, by the way, I ended up spitting out because I could not dissolve it). I assume a good part of it was plant-based materials (cellulose?) so I should be fine. I aborted the knife-eating procedure after around a third of a knife, partly because the dissolving part admittedly take quite a while, and partly because the next round starts soon.

Tournament Round @ 3:00: Interesting idea, except the logistics of this event weren’t very well thought out. Lots of chaos as to organizing the 32 teams takes about 20 minutes to settle down. At 10 minutes per set, that’s a downtime of about 67%, which is pretty bad to be honest. Further rounds take a bit less time to organize, but are still time-consuming; in the end BMT overran its schedule by well over an hour, leading to many of the losing teams leaving early. Not so great on the contest morale in general. Now onto the problems:

First set: #6 is pretty trivial; it is about multidimensional things. Although this is partly due to prior experience; in particular I dabbled in the subject by myself for a few days last year, concluding that the n-cube follows (x+2)^n generation procedure. Thus the “number of 3-cubes that compose a 5-cube” is simply the x^3 coefficient of (x+2)^5, and that is easy enough to compute. [Similarly, (x+1)^(n+1) describes the simplex aka tetrahedron aka triangle, and (2x+1)^n describes the cross polytope aka octahedron, although you do have to fudge the formula a bit to get the component numbers. No other regular polytope generalizes.]

By the way, that ended up degenerating into philosophical discussion on noninteger dimensions, which are technically evaluable by using the gamma function extension of the factorial, and most probably I alienated half of my viewers who go to my school by typing up random arcane math stuff. Although I’m not sure how they managed to survive the earlier portion. Oh well.

The rest of the set is somewhat easy as well, as #5 is a demonstration of the Pythagorean Theorem, and so on. Whoosh!

Second set: Oops, I expended what little energy I had regained during lunch on the first set. Completely fail easy linearity of expectations bash, then timesink on a few earlier problems. We get something like 3 problems, which definitely doesn’t advance, and thus we become another victim of the single elimination system.

That said, the single elimination system sucks on general tournament morale as well; this probably contributes greatly to the evacuation of most of the losing teams; teams like us were only occupied due to the existence of a chessboard.

Chess @ 4:00: After losing rather unceremoniously (we were 4th seed who lost to 13th seed), we decided to play chess. First, we tried to do so in the auditorium on one of the lecture tables, but those tables are seriously small and unstable. That led to the development of a miniature version of chess (4×6). Here’s the setup:

RQKR

PPPP

xxxx

xxxx

pppp

rqkr

Maybe I swapped the queen positions, but that’s rather inconsequential.

Anyways, some rules: 1) Pawns cannot move twice on their first turn for semiobvious reasons 2) no castling 3) no enpassant 4) not sure how pawn promotion works, but since nobody’s ever done so before, that’s a moot point.

That game turned out to be a drawfest. Pawns locked places, queen trade rook trade rook trade pawn trade. Whee, a draw. I mean, I’m vastly incompetent at chess (mostly failing due to unwillingness to study opening theory) and believe that trading pieces is the best thing since sliced bread, and I managed to eke a draw.

Although, An.Zh. managed to win a game! Good job to him!

Later on, we migrate to more stable ground (right outside the auditorium) and we play a game of single-board bughouse. Basically, bughouse with two 4×8 boards. No queens (queens become royal kings) and rooks, bishops, and knights are distributed evenly. I think it was RNKB or RNBK, but I may be mistaken. But anyways, that lasts its course, and Aa.Li. comes up with a new variant:

So basically in this variant, the split board system semi-applies. However, each team controls a single color of pieces (why did I type “colour”?). You can only control pieces on your side though. This leads to a technique called “rook sniping” where you save up both players’ moves, and, in rapid succession, move twice to kill a piece without your opponent being able to react. (Actually, it’s not rook-specific, but rooks are pretty effective with this strategy… although the first piece to use this was a bishop if I recall correctly which took down a queen)

—————————-

Awards @ 6:00: Remember, the awards ceremony was supposed to END at 6:00. Good job to Je.Wu. for not carelessing in individual round, and getting something like 5th! And darn we get 3rd team, getting $50 gift certificates specifically to a particular math course which is severely underlevelled, and likely costs more than $50. Those crafty salespeople! We’d actually be more content with 6th or 7th team, who got USB drives instead. Darn, they get tangible prizes 😦 But anyways, despite the disorganization, all is well.

Then we eat dinner, and go back home on BART again. Much contact ensued meanwhile. Oh, and we barely caught the train/bus with several seconds before it left 😀

 

Wow, that was long, and certainly probably does not justify the “semi-long” designation given in the title of my blog. Oh well.

I might actually expound further on the endurance issue later, but I hope you get somewhat of an idea why I don’t have any of it in the example posted earlier.

Criticism!

Please feel free to criticize me at any point. I do become disillusioned when I’m actually correct you know, when I don’t expect to be correct. This post is written mostly** as an extended response to ellerej’s comments to my post on Language.

(**remark: well, the first part of the post)

I first briefly scanned his/her (If you like, you may clarify on your gender; I will assume that you are a male in the remainder of this post.) blog. Clearly he has forayed into the realm of linguistics much more deeply than I have, with pretty much all the linguistic knowledge I know comes from a rather haphazard fashion of reading random Wikipedia pages* (this extends to most other topics such as history or computer science or Minecraft physics, say). Therefore I will admit that I do not have a rigorous treatment on any of these subjects, and do not know of many terms and jargon used to effectively communicate on the subject. But that isn’t really the intent of posting such things. Of course it’s not going to be published in some linguistic/compsci/Minecraft journal, nor do I expect my thoughts to override other peoples’ beliefs.  I do like thinking, however, and thus I randomly synthesize various materials in my brain to create posts, no matter how erroneous it may be.

(*I try to minimize usage of Wikipedia while writing my posts other than to quickly find examples though, mostly relying on retained information from Wikipedia. Hence potential factual inaccuracies.)

On the other hand, the reader may ask, there must be something I’m good enough at. (Note that this is vastly different from the Rathian “expert”.) In particular, the Rathian expert is one which the general populace believe is well-versed with certain facts. I’m talking about people who actually are well-versed with certain facts (relatively speaking of course; I obviously cannot hope that anyone at our high school to be involved in forefront college research at any one subject.)

No, it is not any of the sciences, hard or soft. I find myself trying to justify many of these concepts to myself to ensure that things are true, and epically fail doing so. (confined to two things: 1) spend as little effort as possible while 2) getting a good grade. You know, MSJ A-range grade.) Yeah, point #1 isn’t particularly conducive to my attempts of justification, as I just go “fuuuuuuu I suck at chem I’m (too lazy to/can’t actually) prove this” or “fuuuuuuuuu this chem concept is too easy and is trivial, chem y u no get harder to prove”. Perfect recipe for defeat, except I’m not actually sad because I already expect the defeat. [That said, I seem to be handling this chemistry thermodynamic stuff unusually well, given how badly I fared when I did thermodynamics in physics. Conditioning? I don’t know.] So basically, I fail at chem. Apparently lots of people are failing worse at chem (gradewise, possibly concept-wise?) though. And they get really good homework packet grades too. [Yeah, homework packets are the reason why my grade isn’t as high as it should be. Let’s just say that I could probably improve about 6-7% with a better homework grade, oops. I also, for better or worse, decline certain inherent advantages given to us in tests, such as scientific calculators (which did not turn out spectacularly, but it could have been worse I guess).]

Well, that’s a lot of tangent for one paragraph. Let’s just say that I suck at writing too.

I guess you could say that I’m competent at math. That may be true. However, rarely can I solve problems that you guys throw at me in due time. I don’t work like that. Most likely, any math problem that you tell me to do will be stuck in a very long queue of things to do, and I will probably defer that problem until you forget about ever having asked the problem. This usually occurs for geometry problems that various people on gmail chat ask, and I will admit that I have a very real deficiency in that subject of math. In particular, my score distribution on the USAMO last year looked something like this: 7/7/0/7/2/7. That’s 28/28 for non-geometry (1,2,4,6) and 2/14 for geometry (3,5). [Or maybe I’m just too lazy to draw a diagram. This is true too.]

However, it is to my shame to announce that I can’t solve number theory on the spot either. This might not bode very well on pretty much any USAMO besides last year’s (Basically, last year was #2 #6 combo, #1 nt, #4 alg if I recall correctly), where the number theory and algebra coaligned with the #1/4 (the first problem of days one and two, i.e. the easiest problems).

Actually now that I think of it, I can do #1/4 geometry (take the USAMO the year before that). But that’s about all the geometry I can do. I guess I could possibly do #2/5 NT/alg as well. Not too stable I’d say.

Hmmm. And then there’s combinatorics, which supposedly is my impenetrable bastion. I guess possibly maybe.

Am I good at math? You decide. Although, to be fair, I’m not planning on making this blog much of a math blog (maybe concepts/olympiad strategy (do I even possess this!?) or something, but most likely not actual math problems), so that point’s moot. Still remember what was the original intention of this post?

 

Hah, I bet you scrolled up for that one.

 

As you can tell, I’m apparently very prone to digressing. And thus I suck at writing. Again.
So basically, we can establish that I am not going to be an expert at whatever topic I am writing my blogpost on [this blog at least]. That much is a given. After all, if I were an expert at a topic, why would I even bother posting new innovative ideas on a measly blog? Well, I can’t really imagine why someone would post something that doesn’t fall under one of the categories:

1) Personal stuff/sentimental/rants. You can’t really replicate this. While your experiences and mine may coalign mostly, I try to present these things in a different light. Not necessarily in a better light, a different one. Oh well, I guess that’s a start.

2) Things semi-independently found. Basically, without having had a formal treatment [challenge: ANY treatment] on a particular subject, talk about it. It’s a very interesting exercise and I encourage more people to partake in such an activity, especially since you can’t dwell in your comfort zone. It’s also a very interesting experiment from an outsider’s point of view as well, seeing perspectives largely unaffected by subject-specific discourse, although I’m not sure how isolated one actually can be from a subject.

3) Statistics, infographics, etc. Eh, this blog will not contain solely of statistics. I do enough of that on a daily basis already. Although, I do realize that some people like blogs filled with that stuff. Your choice I guess.

Hm, I think I covered too much under #1. Oh well.

 

Now that I’ve not-very-clearly outlined exactly the extent of seriousness my posts should be taken to, you may continue to criticize away 😛