「峰不二子という女」/「巖本英利」

So this week’s Lupin III/Fujiko show turned out to be Phantom of the Opera. Sure, operas are classy and this iteration of Lupin III is fairly classy, so why not? Well, it turns out the opera they used is one of the few that I had actually studied (to some degree) back in high school music. The opera is Tosca, by Giacomo Puccini, and is apparently near the top of the list of famous operas. That said, I think it’s safe to say that anime is near the bottom of the list of places I expected to hear an aria or something and make me go “Wait, this sounds awfully familiar”.

Tosca is about a famous singer, Floria Tosca, her lover, a painter by the name of Mario Cavaradossi, and Baron Scarpia, the chief of police and a guy who really, really wants Tosca. Basically, Tosca and Cavaradossi are in love. Cavaradossi also happens to be helping one of his Republican friends escape from the police. Conveniently, these two things make Scarpia hate Cavaradossi a lot. This causes a lot of trouble for the lovers. Spoilers: Tosca ends in tragedy.

There are two pieces that were used in this episode.

Tosca Divina

Part of this piece was performed on stage when the spotlight fell on Ayan. Just like in the episode, Scarpia arrives and is looking for Angelotti, the guy that Cavaradossi helped escape. Obviously, with the spotlight dropping on them, they had to improvise, but the first two lines that they sing kinda don’t make sense. After that, though, is pretty much what happens at the end of that scene.

The scene in context is basically Scarpia tricking Tosca into thinking that Cavaradossi is cheating on her with the lady he’s drawing in the painting. Tosca’s line about being a prisoner of royal celebrations (è prigioniera dei regali tripudi) is the end of a bit about her coming to the church to tell Cavaradossi that they wouldn’t be able to meet that night. Scarpia’s line after that, about his poison working (Già il veleno l’ha rosa!), doesn’t make sense without the context, because it’s about him thinking about how she’s buying into his bullshit about Cavaradossi cheating.

Basically, Tosca is pissed. She talks about how their beautiful nest is soiled with slime (Oh mio bel nido insozzato di fango!) and then basically says she’s going to kill that cheating bitch (so to speak) who isn’t going to have Cavaradossi tonight (Vi piomberò inattesa! Tu non l’avrai stasera, Giuro!). Scarpia goes “In church!” (In chiesa!) because they’re talking in a church and Tosca saying how she’s going to totally murder that lady is probably something she shouldn’t do in a church. That’s okay, replies Tosca, “Dio mi perdona”, or God will forgive her, presumably because the cheaters deserve it.

So Scarpia successfully gets Tosca pissed at Cavaradossi. After the singing from just above, Tosca goes off to find Cavaradossi and Scarpia gets a guy to trail her. This, of course, leads to all sorts of unfortunate things later on. For now, though, Scarpia gets to sing about how much he wants Tosca with the undertone being that he’s super evil and all this happens juxtaposed against a church procession and the act ends.

It’s not obvious in this part of the episode, but there are a few interesting parallels that we can extract from this if we maybe read too much into this. Obviously, Fujiko is Tosca and Zenigata is Scarpia. I’ve never seen much Lupin before, but here, Zenigata seems like a terrible person, what with molesting Fujiko and all. He really seems to despise Lupin, much like Scarpia hates Cavaradossi, and he uses Fujiko to try and get to him. And after this the analogy breaks down, but, oh well.

Vissi d’arte

This is probably the aria that Tosca’s known for. It plays on the record player in the hideout when Ayan is explaining the reveal to Fujiko. At this point, Scarpia has Cavaradossi taken away to prison after having revealed his Republican loyalties. He takes advantage of this situation and offers to free Cavaradossi if Tosca gives herself to him. As he waits for her answer, she starts singing this aria, which is basically her crying out to God asking why she has to go through this.

The opening line is “Vissi d’arte, vissi d’amore”, which is “I lived for art, I lived for love”. It could be fansubber cleverness, but Fujiko outright says that she gets Ayan “lived for love, but…” I think there’s more to the use of this piece in this part of the episode than there was with the first part. Here, Ayan talks about how all she wanted was to be with Darenzo but no one would accept it. Of course, the trial she has to go through isn’t as bad as Tosca’s and definitely ended up much better than Tosca did.

Actual spoilers for Tosca follow if you care about that kind of stuff. Shortly after Tosca finishes Vissi d’arte, Scarpia ups the ante by ordering Cavaradossi to be executed. Buuuuuuut, if Tosca gave herself to him, he’ll make it a mock execution. Tosca ends up agreeing, Scarpia gives the order, then Tosca knifes him. Everything works out, except that Scarpia is a pile of dicks and actually ordered the execution for realsies. Tosca throws herself over an edge in grief and to avoid the guys who just found Scarpia’s body.

「涼宮ハルヒの消失」/「茨乃」

Happy $\pi$ day. Once again, Nadeko will bring us in:

Snowy Mountain Syndrome is the third story in The Rampage of Haruhi Suzumiya, the fifth volume of the light novel. It’s the first story that has yet to be animated. It’s also a story that contains the dread spectre of mathematics.

So our SOS-dan is stuck in a mysterious cabin in the middle of a snowstorm on a mountain. They find a mysterious contraption that has an equation displayed:

$$x-y=(D-1)-z$$

and they are to provide $x$, $y$, and $z$. Koizumi and Kyon are confused, but Haruhi rightly identifies this equation as Euler’s polyhedron formula, which is also very often referred to as just Euler’s formula. If you’re referring to it in context, it doesn’t matter that much, but it’s useful to distinguish between all the other things that Euler discovered, which is a hell of a lot.

First, we should probably go over some basic definitions. When we talk about graphs, we’re not talking about bar graphs or pie charts or the like. We’re also not talking about graphs of polynomials on a cartesian plane or other such functions. Graphs are a mathematical structure which, when drawn, looks like a bunch of circles and lines.

Formally, a graph is a pair $G = (V,E)$ where $V$ is a set of vertices and $E$ is a set of edges. Vertices can be any old thing, but each edge is defined as a pair $(u,v)$ where $u$ and $v$ are vertices in $V$. When we draw graphs, we just draw a vertex as a circle and draw an edge as a line that connects the two vertices it’s defined as.

And that’s it! That’s the most general definition of a graph, which means we can end up with a graph that’s completely empty or a graph that’s just a bunch of vertices with no edges in between them. We can even have multiple edges going in between two vertices. Of course, often times, we’d like to add some more constraints, depending on what we want to do with our graphs. Very often, we’d like to restrict the number of edges between two vertices to one and that’s what we’ll do.

Back to the formula, usually it’s given as $\chi=v-e+f$, where $v$ is the number of vertices, $e$ is the number of edges, $f$ is the number of faces, and $\chi$ is called the Euler characteristic. That makes $x=v$, $y=e$, $f=z$ and $D-1=\chi$. Now, the only thing here that we haven’t seen defined yet is a face. Intuitively, we can see that’s just the space that’s bounded by the edges.

What I find strange is the explanation in the novel that $D$ stands for the dimension of the polyhedra. As far as I know, this only works in the three-dimensional case for platonic solids. Once we generalize the structures to other kinds of polyhedra and topological surfaces, that analogy breaks down.

Anyhow, the way the formula is applied in the book is the use that I’m most familiar with, which is as a property of a planar graph. For planar graphs, $\chi=2$. In the novel, they deduce that $\chi=1$ since $D=2$ and that only works because they didn’t count the large face outside of the edges as a face, which we usually do.

But what is a planar graph? Well, if you go back to our definition of a graph, you might notice that all we’ve done is said that it’s a bunch of vertices and edges. We’ve said nothing about how to draw a graph. Usually, we represent vertices as circles and edges as lines in between those circles, but other than that, there’s really nothing telling you what order to draw your circles in or whether your lines have to be completely straight or not or how far apart everything has to be. How you choose to represent your graph is up to you, although if you draw your graph weirdly, you might make the people trying to read it angry.

Informally, a planar graph is a graph that you can draw with none of the edges crossing each other. This seems like a kind of silly thing to be worried about, because it seems like you could just keep on drawing a graph until it works out. Well, for edges with a lot of vertices and edges, it’s not obvious and even for really small graphs. For instance:

At a glance, it doesn’t look like the drawing on the right is planar, but all we have to do is drag one of the vertices into the middle to get the drawing on the left and it turns out they’re both the same graph, $K_4$, the complete graph of order 4.

That’s where Euler’s formula comes in really handy. It gives us a way of figuring out whether or not our graph is planar or not without having to fiddle around with placing edges properly and stuff. You already know how many vertices and edges you’ve got, so all you need to do is make sure you’ve got the right number of faces.

So it’s probably pretty clear at this point that you can’t draw every graph without the edges crossing. We can say something interesting about those graphs too, which just turns out to be another characterization of planar graphs, but oh well. But first, we have to introduce the concept of graph minors.

Suppose we have a graph $G=(V,E)$ and an edge $e=(u,v) \in E(G)$. If we contract the edge $e$, we essentially merge the two vertices into a new vertex, let’s call it $w$, and every edge that had an endpoint at $u$ or $v$ now has $w$ as the corresponding endpoint. Then a graph $H$ is a graph minor of $G$ if we can delete and contract a bunch of edges in $G$ to get $H$ (or a graph that’s isomorphic to $H$).

It turns out that every non-planar graph can be reduced to a minor of one of two graphs. The first is $K_5$, the complete graph of order 5:

The second is $K_{3,3}$, the complete bipartite graph 3,3:

These two graphs are the smallest non-planar graphs, otherwise we’d be able to reduce them further to get another non-planar graph. Like I mentioned before, this is a characterization for planar graphs too, since a planar graph can’t contain a $K_5$ or $K_{3,3}$ minor.

I guess I’ll end by saying that graphs are hella useful, especially in computer science. A lot of people complain about never using math like calc ever. If you’re a developer, you’ll run into graphs everywhere. It’s pretty amazing how many structures and concepts can be represented by a bunch of circles and lines.

A visual novel about disabled girls with origins from 4chan based on some extra pages in some doujin sounds like the most horrible thing and shouldn’t exist, but it exists and is actually not horrible at all. But everyone knew that already when the demo came out, what, three or four years ago? Everyone was waiting for it, but kind of like Starcraft II, I don’t think anyone expected it to get released within our lifetime.

So, the story goes the main character, Hisao, gets a heart attack and discovers he has a heart condition which leads to him being enrolled in a high school for students with disabilities. And then life becomes an adventure with choices.

Katawa Shoujo is a galge and seeing as how I played it right after I finished Little Busters, it was really hard not to compare it to Key’s brand of visual novels. I’d talked about Little Busters before and mentioned that it seemed more ridiculous than the usual Key stuff. Instead, Katawa Shoujo was able to get me to care about the characters more, simply because what they were going through was a real thing that could happen.

And because these were real things, the characters felt more like real people. A big reason why moe template characters don’t feel like real people is because the only flaw that they have is whatever otherworldly problem manifests soon after they meet you. Real and serious problems make the characters more believable and it seems like the writers really had to think about the implications of what each disability meant for the development each character.

For instance, one of the more interesting characters in Little Busters was Kud, whose problem was basically being half-Japanese and half-foreign and not knowing where she belonged. Of course, her Engrish was played up for the uguu factor occasionally, but her being unable to find belonging made her much more interesting than other uguu~s.

Realistic problems affect the characters. Some of them are born with them. They act differently than those who weren’t. Some of them overcompensate for what they don’t have. Some of them don’t care. In some ways, it’s not so much the disability affecting the character, but how the character’s personality is magnified by how they deal with it.

The characters have very strong and distinct personalities. Normally, the main character is a blank stand-in so that you can feel like that guy, but Hisao has just enough written about him that he’s his own character. At the same time, he’s still undefined enough that his personality changes depending on which girl’s route you’re on. A result of this is that in every route, he has really good chemistry with the heroine.

So since the individual routes matter and hit upon different things, let’s go through them.

Shizune’s route was the least enjoyable. I’ll admit that a large part of why I didn’t like Shizune’s route is because I didn’t like Shizune or Misha. They’re interesting when they’re together, but I didn’t like either of them because I don’t think I’d get along with them. I’m not really into the short hair or megane either. But most importantly, I thought most of the route was kind of boring, probably because the problem in this route was relationship drama.

Lilly’s route was enjoyable. Like Shizune, Lilly’s had her disability since birth, so she’s used to it by now and none of her problems really come out of that. And in fact, she’s a well-adjusted person who’s doing well, so she’s not having any problems on that front either. Most of the route is actually about Hisao dealing with his new life and how that affects his relationship with Lilly.

Emi’s route was one that I didn’t expect to like as much as I did. I don’t usually like the archetype that Emi’s character is drawn from, but I think her trolling with the nurse made her more likeable than just the energetic girl. She’s one of the characters who wasn’t born with her disability and has it play a role in the story (the other being Hanako). While the theme in both of their stories is kind of the same, their stories are different in the way they handle their situation. Emi is pretty conscious about her situation and does stuff about it, whether or not it’s actually helpful. Also, the nurse seemed eerily similar to Souma from Working!! (and by transitivity, Izaya from DRRR!!).

Rin’s route was the most surprising. In the other routes, Rin’s disconnectedness and resulting lines are pretty amusing, so I went into it kind of expecting some eccentricity and laughs. It turned out to be rather serious and not hilarious at all. It definitely colours her responses, even outside of her route. Like Lilly, her disability isn’t the root of any problems she’s facing. I can’t recall off the top of my head of any other visual novels where the protagonist and one of the heroines have such unpleasant development and struggle to understand each other.

Hanako is my favourite character, if not necessarily my favourite route. Out of all the characters, I think Hanako’s backstory and story are the most Key-like. She’s basically the shy, quiet girl with tragic past (so the opposite of Emi) and most of the route involves protagonist-kun breaking her out of her shell. Well, at least until the end, where the resolution to her route is not very Key-like at all. While the ending was good, it seemed kinda short. Or I don’t know, maybe I just wanted to see Hanako be happy some more.

Obviously, Katawa Shoujo isn’t about disabilities or disabled people or how disabled people are people just like us. I’d like to think that we don’t need a visual novel to teach us that. And, I mean, the central problems to the stories have very little to do with the characters’ disabilities. It’s all about Hisao learning to understand people, especially the girls he’s building relationships with. The caveat is that they, and most people, make that very hard to do.

She doesn't seem that excited

So last time, Tetra was being enlightened by MC-kun about definitions. This actually arises from MC-kun using prime numbers as a motivating example.

Primes are megas important in mathematics and even more important today. The entire branch of mathematics called number theory is all about studying the properties of prime numbers. They’re so useful that we’ve done stuff like extend the notion of prime elements to algebraic structures called rings or apply analytic techniques to learn more about them, but we’ll stick with elementary number theory for now.

Now, for hundreds of years, we’d been studying number theory only because it’s cool and mathematicians love prime numbers. Last time, I mentioned some examples of math preceding useful applications. Well, number theory is a really good example of that, because in the 70s, we found a use for it, which is its main use today, in cryptography. There have been some new techniques using some algebra as well, but for the most part, modern cryptography relies on the hardness of factoring primes. Neat!

Okay, so we’re back to the original question that MC-kun tries to get Tetra to answer, which is, what is a prime number?

Definition. An integer $p$ is prime if and only if $p\geq 2$ and the only positive divisors of $p$ are 1 and itself.

MC-kun explains that the motivation for excluding 1 from the definition of a prime number is because we want to be able to say that we can write every number as a unique product of prime numbers. This is very useful, because now we know we can break down every number like this and we can tell them apart because they’re guaranteed to have a unique representation. This is called unique prime factorization.

Theorem. Let $a > 0$ be an integer. Then we can write $a = p_1p_2\cdots p_k$ for some primes $p_1,\dots,p_k$. This representation is unique up to changing the order of terms.

We can show this by induction on $a$. We’ve got $a=2$ so that’s pretty obvious. So let’s say that every integer $k\lt a$ can be decomposed like this and suppose we can’t decompose $a$ into prime numbers, assuming $a$ itself isn’t already a prime since it would just be its own prime decomposition. Then we can factor $a=cd$ for some integers $c$ and $d$. But both $c$ and $d$ are less than $a$, which means they can be written as a product of primes, so we just split them up into their primes and multiply them all together to get $a$. Tada.

As a sort of side note, I mentioned before that primes are so useful that we wanted to be able to extend the idea of prime elements into rings. Well, it turns out for certain rings, it isn’t necessarily true that numbers will always have a unique representation when decomposed into primes. This is something that comes up in algebraic number theory, which is named so because it involves algebraic structures and techniques. This was invented while we were trying to figure out if Fermat’s Last Theorem was actually true (which needed this and other fun mathematical inventions from the last century that implies that Fermat was full of shit when he said he had a proof).

So at the end of the chapter, after Tetra gets her chair kicked over by the megane math girl, we’re treated to a note that acts as a sort of coda to the chapter that mentions that there are infinitely many primes. How do we know this?

Suppose that there are only finitely many primes. Then we can just list all of the prime numbers, like on Wikipedia or something. So we’ve got our list of primes $p_1,p_2,\dots,p_k$. So let’s make a number like $N=1+p_1\cdots p_k$. Well, that number is just a regular old number, so we can break it down into its prime factors. We already know all the primes, so it has to be divisible by one of them, let’s say $p_i$.

Now we want to consider the greatest common divisor of the two numbers, which is just the largest number that divides both of them. We’ll denote this by $\gcd(a,b)$. So since $p_i$ is a factor of $N$, we’ve got $\gcd(N,p_i)=p_i$. But then that gives us $p_i=\gcd(N,p_i)=\gcd(p_i,1)=1$ by a lemma that says that for $a=qb+r$, we have $\gcd(a,b)=\gcd(b,r)$. This means that we have $p_i=1$, which is a contradiction, since 1 isn’t a prime number, and so I guess there are actually infinitely many primes.

So the nice thing is that we won’t run out of prime numbers anytime soon, which is very useful because as we get more and more computing power, we’ll have to increase the size of the keys we use in our cryptosystems. Luckily, because factoring is so hard, we don’t need to increase that size very much before we’re safe for a while. Or at least until we develop practical quantum computers.

「正月２０１２」/「ZEN」

So my predictions have been a bit off, but there’s still plenty of time for Little Busters to get a KyoAni anime on that timeline! What I think is more surprising (other than Haruhi getting more anime before LB) is that the translation for the visual novel is finished and I’ve played through it, long before a Little Busters anime has even been announced.

Little Busters is an interesting experience for me, because it’s the first “real” Key visual novel I’ve played without knowing much going into it. Sure, there’s Planetarian, but that’s relatively short, so I don’t count it. Sure, there’s Angel Beats, but that’s not a visual novel. And sure, I’ve played Clannad, but it’s Clannad and I know everything about Clannad.

I mentioned before that Angel Beats made me wonder whether I really like Key or if I just really liked Clannad. Even better than an anime, I think the Little Busters visual novel is a perfect opportunity to see where my tastes lie.

Like any good Key work, Little Busters has to have a theme. That theme happens to be adolescence or childhood. Alright, then. From this, there are a bunch of things that are pretty similar to Angel Beats. We’ve got the setting down and there’s a good chunk of the game that’s spent on trying to put together a baseball team. At a glance it seems like it’s all about living out your youth and all that. The common route mostly just made me wonder why they bothered to create Angel Beats when they had this lying around.

You’ve got your usual suspects in the cast: socially awkward childhood friend who likes cats, disgustingly cheerful nice girl, shit-stirring genki girl, suspiciously combat-hardened and cool onee-sama, quiet book girl, and dojikko with verbal tic. But, the main character isn’t the usual Key template blank but mildly snarky dude. Instead, you’re a Hayate (from Hayate the Combat Butler) except you’re kind of weak instead of absurdly competent. You’ve also got a bunch of childhood friend bros, the Little Busters, who watch your back and are actually pretty important to the main story. Obviously, every important character ends up on the Little Busters baseball team.

Structurally, the whole thing is pretty similar to Clannad. You’ve got all of your routes that you have to do before you get a swing at the route that ties everything together. What’s different is the common route, where you’re building up stats and rounding up people and comedy happens. I actually like the common route, if I ignore being put through it about six times.

Where I’m pretty dissatisfied is with the side routes. I went in expecting the usual Key stuff with fatal sickness and astral projections. I think the main problem with this stuff in Little Busters is that the writers realized that they couldn’t fall back on the same old stuff again, so they tried to spin up some new awful tragedy for each character.

Before, the tragedies were pretty grounded. Someone lost a family member or someone is terminally ill. That stuff is easy to empathize with. The most outlandish stuff is the astral projection or animal spirit stuff, but even then, that stuff is sort of left to mystery.

In Little Busters, they take something simple and try to add another layer to it to try to make it new. So someone loses a family member, but they also regress into a catatonic state whenever they remember. Or someone is feeling out of place because they’re half-Japanese and struggling with their cultural identity, which is a real thing and you can empathize with that. But then they add this crazy backstory about their homeland under civil unrest and it’s like what.

And it’s not like they succeeded in making these developments new. I’ve watched and played almost all of the Key anime and visual novels and that basically let me SEE THE ENDING, so to speak (not that they weren’t making it extremely obvious). When I didn’t predict how a route would go when I got halfway through it, it was because there was the aforementioned ridiculous thing that was bolted on.

The “real” story, as in the right girl’s path together with the final route, is better in that the twists were actually kind of interesting instead of dumb and it’s where it differentiates itself from Angel Beats. How the story unfolds is a bit more clever than Clannad’s handling of the After Story route.

It’s definitely not as great as Clannad and I don’t think even the main route came together all that well. Even though it was better and actually interesting, a lot of it was still kind of ridiculous. I’ll let light orbs go, but this was kind of pushing it.

This all makes me kind of worried about Rewrite, but that has a trailer where a guy fights a dinosaur, so who knows?

He will use this knowledge in less than 24 hours.

So back when I was moving in at the start of the school year, I decided to try out my BD player on my hot new Dell monitor. Since I didn’t want to take out my Kara no Kyoukai BDs from their box, I opted to crack open the cheap flimsy Summer Wars case and throw in the disc. I hadn’t actually tried playing it yet anyway. I opted to skip the OZ introduction and go straight to the opening. First of all, I was disappointed that Funimation set the credits in goddamn Arial. Seriously, what? I find it difficult to believe they didn’t have Helvetica already from one of countless projects they could’ve used it on already.

More importantly, Funimation translated the chapter title of the book that Kenji was reading on the train. It turns out he was reading about Shor’s Algorithm. Shor’s Algorithm is the quantum algorithm that can factor integers in polynomial time. I alluded to it when I talked about Summer Wars and cryptography a while back and now’s as good a time as ever to get into it, since, you know, this means Kenji really is a quantum computer and all.

In quantum computing, instead of dealing with bits that are 1s and 0s, we deal with quantum bits, or qubits, that are denoted $|1\rangle$ and $|0\rangle$. These are really just quantum states. Quantum states have interesting properties, the most famous of which is being in superposition. Essentially, what this means is that we can have qubits that are in a state where it represents 1 and 0 simultaneously.

This turns out to be a very neat property to have. If we’ve got $n$ qubits, then we’ve got like $2^n$ inputs simultaneously. That’s great, right? We can compute things exponentially fast now! Well, it turns out it isn’t that simple.

Suppose that we do end up running an algorithm or something on our $2^n$ states. What happens when we want an answer? Well, we measure the state. The problem with that is that on measuring a quantum state, you won’t get all $2^n$ results. You’ll only get one of those results with a certain probability. What’s more is that once you measure a quantum state, you’re stuck with whatever it gave you and you can’t go back and check for another result.

That is obviously no good, because we want not just any answer, but, you know, the right answer. So it looks like approaching our problem from the naive interpretation doesn’t seem to work. And in fact, that immediately tells us that quantum computers aren’t a magic bullet that can help us solve SAT in a reasonable amount of time.

What we can do is manipulate our resultant quantum state so that the actual correct result we want is the one with very, very high probability and reduce the probability of getting any of the other results to near-zero. Of course, the problem is that we don’t know which result is the correct one to begin with. It’s actually this exact hitch that prevents us from efficiently solving general NP-complete problems.

Luckily, if we want to factor some numbers, we can take advantage of this. We can reduce the problem of factoring any old number into prime factors to the problem of order finding, which I’d mentioned before. For an integer $N$, there’s another integer $m$ such that $k^m=1\bmod N$ and we call $m$ the order of $k$ in the integers modulo $N$.

So suppose we have $k\bmod N$ with order $m$ and we decide to keep on taking powers of it, like $k\bmod N,k^2\bmod N, k^3\bmod N$ and so on. Eventually, we’ll get up to $k^m\bmod N$, at which point, that’s just $1$ and we start our cycle over again. The thing to note here is that this sequence is finite and periodic, which narrows down what we have to search through.

So what do we do? We create a quantum state for our period that’s in a superposition over all the powers of $k$ in $\bmod N$. We don’t know what our period is (that’s what we’re trying to find), so we just choose a large enough one that our order will be in the period. Now, in order to find this order, we do some magic with a Quantum Fourier Transform.

What?

So I mentioned at the beginning that when we look at a quantum state in superposition, we can think of it as returning one of the possible states with a certain amount of probability. How the probabilities for these quantum states work is actually more complicated and it’s from this that we can massage an answer out of our magic quantum algorithm.

Quantum states have something called amplitudes, which are kind of like probabilities, except that amplitudes can be basically any complex number as long as all of the amplitudes $a_i$ for a quantum state satisfy $\sum|a_i|^2=1$. If we’re just measuring a quantum state, this really doesn’t change anything, we just get state $|i\rangle$ back with probability $|a_i|^2$. But when we’re manipulating quantum states, there’s more to the amplitudes to consider.

If we think about amplitudes in terms of waves, we can see that we can cancel waves out. For instance, the states $\alpha|1\rangle$ and $-\alpha|1\rangle$ have the same probability if we were to measure them (since $|\alpha|^2=|-\alpha|^2$), but if we put them together and add them, they’d cancel each other out. What’s more is that since amplitudes are complex numbers, so we can think of them as having a direction (like a vector on the complex plane) and we end up with weird stuff like having things partially cancel out.

So just like a regular Fourier transform extracts some information about frequencies from a sequence, we can use the Quantum Fourier Transform to extract some information about all of this amplitude stuff that we can’t see just by measuring the state. And this is exactly how we can manipulate our state so that it increases the probability of the answer we want when we go to measure it and depress the probabilities of all the other answers.

Now after rambling about amplitudes and Fourier transforms and orders, we should probably step back and see what we’ve done in perspective. What we have is a way to find the order $m$ of an element $k \bmod N$. Well, actually, we don’t even have that quite yet. What we get out of all of that quantum stuff is just some information about $m$ that we have to manipulate further, except we just do that with plain old number theory instead of quantum computation. But, what we’ve kind of stumbled through is kind of a vague description of Shor’s algorithm.

Like I mentioned in the first Summer Wars post, this is the only way we know of so far that lets us factor large numbers (especially the ones that are used in cryptography) within a feasible amount of time. Again, Kenji (or at least his brain) is basically a quantum computer. Coincidentally, I played around a bit with what the implications could be if our brains were quantum computers in an earlier post on Kaiba.

But even if Kenji isn’t a quantum computer, he’s goddamn smart for a high school student. The theory behind the computing part of quantum computing is really just a ton of linear algebra, so an understanding of quantum mechanics isn’t really necessary (which is great, because I don’t know any of that crap). The quantum computing course that I took in my undergrad only had second year linear algebra as a formal prerequisite, but that was mainly to allow students from different departments to take it and also implies that you’ve got first and second year math down at the very least.

「魔法少女まどか☆マギカ」/「ゆーげん」

This show was made for me.

I think that should be your sign that you won’t find a paean to the greatness of Madoka and why it is man’s greatest work.

I mean, I was sold on the show already back when it was just “Shinbo makes an original anime”. That’s all it took.

And then it became “Shinbo makes an original magical girl anime” and I’m like, whoa, even better. I mean, I liked Nanoha. More of that? Yes, please.

And then it became “Shinbo makes an original magical girl anime with character designs by Ume Aoki, music by Yuki Kajiura, and script by Urobuchi Gen”. By this point I’d gone nuts. Magical girls! Designed by Ume-tentei! I mean, yeah, Kajiura and Urobuchi were kinda weird choices for a wide face magical girl show, but still, they’re great too. This will be amazing!

And then the characters! The character designs! Aoi Yuuki! Friggin Chiwa Saitou voicing mahou shoujo Senjougahara. Mizuhashi Kaori voicing a magical girl WITH GUNS. KitaEri voicing swords girl. An adorable mascot character!

If the anime that we got was the anime that was presented through only these CMs, I would have been ready to call it anime of the year already.

And then the actual first episode happened. I watched most of the episode going THIS IS AMAZING and then we get to the witch barrier.

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

And then Mami comes with her Gate of Babylon and her GUNS.

I mean, you can probably imagine my mental state as I wait for each week of this cartoon. And so this goes on and I flip out each week and once episode 3 rolls around I go, screw this, I’ll just watch the live stream.

Yes, I watched episode 3 raw when it aired.

FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF

And of course, each week while I’m watching this, I just send my friend a series of FFFFFFFFFFFFFFFFFFFFs.

And then we get to episode 10. I don’t know if it was apparent, but my favourite character is Homura, probably largely because Senjougahara is one of my favouritest characters. So yeah, we get to episode 10 and this is the best anime of the year and possibly of all years.

The end.

Yes, I know a lot of people have problems with it and a lot of what people have said about the show is pretty hyperbolic. Maybe there it does strain itself in trying to be deep and all. Maybe the characters weren’t amazing. I will admit that it very likely is not the best anime ever.

But I don’t think there’s been an anime that has been quite so attuned to my brain. And that is why I love it so.

「little world」/「momoc」

Yeah, I made the same joke last year. But who would’ve thought that we’d get two Utena derived shows within a year? Certainly not me.

But yes, Mawaru Penguindrum is a show that has penguins in it.

It’s fascinating how different Penguindrum is from Star Driver but still manage to be so heavily evocative of Utena. Having Penguindrum as a comparison, I realized that there really was no reason to expect that Star Driver would make as many callbacks to Utena as it does. I mean, it’s not like Igarashi and Enokido hadn’t worked on stuff together before.

Like Star Driver, Penguindrum exudes its own sense of style. In Star Driver, a lot of that is constrained to the more fantastical parts of the show, like whenever we’re dealing with Cybodies and the Glittering Crux. There’s no such separation in Penguindrum. Even in the real world, the architecture and shots of the locations are still really surreal. So while in Star Driver, there’s a clear delineation between real and fantasy, we’re often left wondering which is which in Penguindrum.

And the penguins don’t help very much in this department either.

Looking back at Star Driver, there actually isn’t that much weird stuff going on during the school life section of each episode. I mean, yes, weird events happen and all, but what you saw was pretty much what happened. In Utena, weird things where we’re not really sure how to take happen in the school life parts, which is the route that Penguindrum takes.

Structurally, Star Driver resembles Utena much more because of the clearly defined events that take place in each episode. On occasion, we’ll be missing, say, a song or something but for the most part, we know how each episode goes down. Penguindrum is not very repetitive at all, at least not in how each episode goes. Yeah, there’s SEIZON SENRYAKU, but those are really more diversions than a shadow play and then ZETTAI UNMEI MOKUSHIROKU followed by a substantial duel or APPRIVOISE followed by GINGA BISHOUNEN and a substantial duel.

Of course, all three still make use of visual symbols and phrases and keep on repeating them a lot. The obvious messages are pretty obvious because they’re shoved in front of you, but it just sort of piles on to you to the point that there are probably some other messages hidden under this mound of symbols you’ve suddenly got. I mean, every goddamn thing that shows up on the screen or uttered by anyone is probably a symbol. Has anyone dissected the Triple-H/ARB songs yet?

Penguindrum also differs from Utena and Star Driver by not being about a prince who strolls into town and upsets the establishment trying to revolutionize the world. Certainly, we’ve got our shadowy cabal trying to revolutionize the world, but it’s really more about two brothers trying to save their sister, which is much less grand. Of course, the penguins would probably deflate any sense of awe pretty quickly.

Not that I don’t appreciate the penguins. I think they’re an awesome part of the show. It’s an interesting way of adding another layer to all of those symbols we’re buried under, in that they supposedly are reflective of the state of mind of the characters they shadow. More importantly, their antics are just plain enjoyable and the show would be much more depressing without them.

Does the choice of penguins mean anything or does Ikuhara just really, really like them? It is a mystery.

Man, I wish I had a penguin that mirrored my internal thoughts and did errands for me and was invisible to everyone except me.

「FF系　マブラヴタイトル」/「ぽかり＠ぴくしぶ」

Muv-Luv is a journey.

Maybe saying it’s the longest visual novel I’ve played is a bit unfair because it’s really two or three games, depending on how you see Extra and Unlimited, and there was a good two or three years until Alternative was released. Still, going through the entire thing takes a ton of time.

The way I like to think about Muv-Luv’s unique structure is by comparing it to Clannad, where the really good stuff, Alternative and After Story, requires a lot of time invested beforehand in the content that comes before it. That’s not to say that the non-Alt/AS stuff is bad, but it’s definitely not earthshattering. And in the case of the Clannad visual novel, it’s probably not as important.

This isn’t true for Muv-Luv. If you’re only in Muv-Luv for the sci-fi, going through the school life hijinks of Extra is going to seem torturous but it is absolutely vital. This isn’t the same as Clannad’s school life routes or Fate/stay night’s Fate route. Extra matters, possibly even more than Unlimited does.

So when I say school life, I mean Muv-Luv Extra is basically your standard school life harem thing. Unremarkable guy goes to school and all of the girls he knows is inexplicably attracted to him. You’ve got childhood friend, mysterious rich transfer student, kuudere, and class president. Other characters include a bro and two teachers. And comedic and romantic things happen in this part and by the end of it, you’ll have a girlfriend. Congratulations!

So you finish Extra and suddenly the title screen changes and all of the heroines are wearing different uniforms. Here is where I’d really like to have been around for when this thing first came out because I have no idea if anyone expected this and what the reaction was. Even more so once you start a new game and find something called Unlimited. Now in Muv-Luv Unlimited, we start with some scenes from Extra. See, because you’re the same guy from Extra. Except now you wake up, walk outside your house and find that the city has been destroyed. HMMMMMMMMMMMMM.

Essentially, this entire thing is about Shirogane Takeru, a normal guy who gets thrown in to an alternate universe in which humanity is under attack by aliens and are losing. How does he deal with this? By breaking into a military base to steal a mecha and save the day, of course. Except that he can’t because he’s a high school student from 2000s Japan and doesn’t know about anything and knows no one.

That last thing pretty much guarantees he’s dead, but with some luck, he ends up becoming a trainee at the local UN military base and he gets a chance to show off his chops. Except, again, he’s a high school student so he fails miserably and holds his entire squad back because of his ineptitude. Unlimited is basically him learning the ropes and somehow getting it. This part ends fairly uneventfully, which brings us to Alternative.

In Alternative, Takeru mysteriously starts over at the beginning of where he was in Unlimited, except he’s retained all of his memories and experiences from Unlimited. This time, he’s going to do it right, which, as it turns out, is quite difficult even if he’s not the complete failure from before. It turns out saving the humanity is hard!

Here’s where everything exciting happens. Political intrigue! Mecha combat! Alien horrors! Military briefings! We finally get to see the mecha in action and there are some really fantastic action sequences. Yes, this is a visual novel and yes, it relies on tricks similar to Fate/stay night that use sprites and the visual novel engine to create a sense of dynamism to the combat. The other thing that adds to it is how the characters work together. Good squad combat is not something you see a lot of in mecha anime, but it’s here and I’d say it’s plays a pretty big part in the story’s themes.

This is also where all of the emotional payoff (read: gutpunches) is. And this is where all of the time you spent with Extra comes in, as the nature of the world you’re in is revealed and casualties mount. While Unlimited and Alternative take place in the same world, Unlimited doesn’t have quite the sense of danger that Alternative does because the aliens do not mess around when they show up. Sometimes we kind of forget that there’s a reason that humanity’s losing. All of this causes a number of oh, shit moments.

As we move from Unlimited to Alternative, the goal changes from trying to get the hell out of crazy apocalyptic world and get back to fun times high school to trying to save that world and the people in it. So now, he’s invested in that place, except that’s a scary place to have people to care for, especially if they’re fighting aliens bent on their destruction. What’s more is that even if he does end up finding a way to gtfo, can he bring himself to abandon everyone so he can chillax with his harem back in Extra?

Muv-Luv is not really about saving the world. It’s a lot more personal than that. How else would you explain Extra? It’s about Takeru coming to terms with this incredible situation he’s been thrown into and rising to the challenge of dealing with it. And it’s something that’s true for all the characters in Alternative. Everyone has things they would rather be doing, but instead, they have to deal with this terrible world filled with aliens and loss and it’s up to them to deal with it and decide to do something about it.

The game’s genre is “a tale of love and courage” for good reason.

「現象数式使い」/「あや波平」

I don’t remember where I heard about Sekien no Inganock, but when I was trying to figure out which visual novel to play next, Inganock jumped out because of how different its premise was. I mean, the last few visual novels I read before it were Cross Channel, Tsukihime, Sharin no Kuni, and Muv-Luv and all of those started off with high school shenanigans and even if they did end up in very different places, all of the principal characters were a guy in high school and other high school students.

Sekien no Inganock is set in a steampunk city inhabited by half-human, half-animal people and is about a travelling doctor. He goes around healing people with, uh, math, I guess? So you can tell that I like this guy a lot already. Besides that, he’s fairly calm and unmoved and his brand of snark is pretty deadpan.

Anyhow, I really love everything about this visual novel. Yeah, the story is kind of obtuse, especially once it gets close to the end. And yeah, that internal monologue system is convoluted and impossible to beat without a walkthrough. But everything else? Fantastic.

The art is stunningly gorgeous, even the character sprites. All of the landscapes are great, but it’s the monster event CGs that are amazing. It’s one of those games that I probably don’t want to see as an anime because an anime will never be able to capture the art (kind of like how the Steins;Gate anime can’t retain huke’s texturing). The music is really fantastic. I really liked the voices, whenever they were present. All of this adds to the great atmosphere and setting. This is something that’s shared amongst all of the games in the What a Beautiful series and this alone is enough to get me to pounce on the rest of them if I ever get the chance.

What separated Inganock from Sharnoth was the characters. I already mentioned our travelling doctor main character Gii. He’s great. But the other character that I thought was awesome was Ati. She’s a catgirl tsundere bro, kind of like Ami from Toradora. Regarding Gii, she waffles between the line of friendship and romance. They help each other out in various business ventures and get drinks at the pub. She’s more of the street smart one, since Gii is kind of a nerd, being a doctor and all. These two really made the game for me and when I got to the end of Ati’s story, well, I mad.

A lot of people complain about the repetitiveness of the story. I guess that aspect of it reminded me of Star Driver because it wasn’t too long after it had finished and we all remember all of the people complaining about how Takuto always wins. Well, it’s the same thing here. We get an encounter and Gii figures the monster out, so he stretches out his right hand. And then after Porshion, who we are assured is not human, burns the monster up, we cut to some dudes with a clock or watch trying to climb some stairs. This is the sort of stuff I enjoy.

I didn’t go in with the expectation that it would answer every question I had and I didn’t really have a desire to understand everything I didn’t get. It was just a really nice thing to experience and I was quite satisfied with having gone through it once I got to the end of it. Well, not quite satisfied in that I’ll be jumping at every bit of news of more WAB games getting translated.

「うさぎドロップ」/「ゆーた」

Single dad manga? Why not? Yotsuba&!’s great!

I don’t know whether there’s much I can add about Usagi Drop. The whole thing is fairly straightforward. It’s a great story about a guy who ends up having to look after a kid all of a sudden and gets into the ups and downs and details of it. I’m sure we’ve all been told that raising a kid is hard and doubly so if you’re the only doing it. Although I guess Daikichi lucked the hell out because Rin is probably the best kid you could hope for.

Okay, maybe I can say something about the much more contentious second half of the story, in which Rin is no longer an adorable child but is a teenager. The immediate reaction is “NOOOO why can’t Usagi Drop be about Daikichi taking care of little Rin forever?” but you know, it doesn’t work like that. Kids grow up and parents should get a chance to feel proud if their kids turn out to be decent human beings, which Rin is in spades.

But also important is that it has some loose threads that need to be resolved. Does anything end up happening between Rin and her mother? Do Daikichi and Kouki’s mom get together? Or do Rin and Kouki get together? I’d imagine a ton of people were interested in seeing how the last two questions get answered. Which of course leads us to the ending.

I’ve already talked about the ending and I’ve tried to understand where it came from. So a lot of people are pissed at the ending because it’s weird as hell and came out of nowhere. But more importantly, it was basically a punch in the gut for everyone who was waiting on those last two questions because it basically resolved the “problem” (because if you think about it, you can’t really have both happen) in the worst way imaginable.

Really, that’s the only snag of the entire thing and I’ll admit it’s a pretty big one. One of the great things about Usagi Drop is its realism and believability, which this development was not. Of course, this is only a concern if you’re at all interested in the manga. If your only desire is to remain in bliss solely in the realm of a guy tries to raise a kid without knowing how, then stick to the anime and the first half of the manga, which is the most adorable thing.

「Mage Killer」/「三輪」

The first time I read Fate/Zero, I got through about half of the prologue before forgetting about it. That was before I read Fate/stay night.

One of the great things about Fate/Zero is that the particular Holy Grail War it covers is serious business. Instead of having a bunch of high school students kind of flail about, you have some powerful magi all scheming against each other. These dudes know what they’re doing. And actually, that all of these guys basically off each other one way or another kind of explains why we’re left with high school students ten years later.

The main draw of Fate/zero for me has always been that I’d heard it’s particularly brutal and that there was a magus going around ruining everyone with the help of modern technology. After reading Fate/stay night, Kiritsugu’s character became much, much more interesting. As much as people seem to think that Rider is the best (and I’ll admit he is pretty fantastic), Kiritsugu has always been the most compelling character to me.

Those who are familiar with Fate/ will know that it’s all about dealing with ideals, whether it’s defending your ideals, sorting out your ideals, or having your ideals challenged. It’s kind of an easy subject to bring up when fighting for the Holy Grail. Fate/stay night is essentially about Shirou sorting out his ideals. In Fate/zero, we have Kiritsugu, who has the same ideals as Shirou, but ends up choosing a vastly different way of realizing them.

This combined with the high calibre of opponents means that there is a ton of cool stuff that goes down in this story. The fact that these are people prepared for the Grail War and not high school students not only means that their fights are better just because they’re better, but they’re also much better prepared. Calling the Grail War a war in Fate/stay night always seemed a bit silly to me, but in Fate/zero, I think there are enough casualties and heavy weaponry to justify it.

Of course, that’s all after I went through Fate/stay night. So what drove me to try and read the books the first time? Well, I’d just finished playing Saya no Uta at the time. Saya no Uta probably remains one of the most horrifying and disturbing things I’ve read. So obviously, I was looking for more in that vein and found this light novel that the same guy wrote.

On the physical horror front, I think Team Caster’s got it covered. In terms of emotional despair, we’ve got everyone else who’s connected to the events of Fate/stay night. It’s mentioned somewhere in the notes of the first volume that there is basically no happy ending for Fate/zero given what we know plays out in Fate/stay night. Basically, everyone involved bets the house on winning the Grail and ends up ruining their childrens’ lives in the process.

It’s going to be a long three months once the first half of Fate/zero is finished. I guess there’s uncensored Team Caster fun times to look forward to in March, though.

「シュタインズゲート・瓦礫と助手」/「huke」

Even though the track record for anime adaptations of Nitro+ games has been abysmal, I couldn’t help but hope that Steins;Gate would turn out well. After all, it’s the best visual novel ever or something according to Japan. And if it didn’t pan out, I guess I’d just fall back on the visual novel with huke’s pretty art.

And as I was watching it, I thought it was pretty funny and all. I was still trying to figure out why Japan loved this thing so much. It wasn’t until someone mentioned something about 2ch jokes that I think I got it.

So I’d known that Steins;Gate was about time travel, based on the synopsis. What I didn’t know was that the story was basically based on the John Titor story. And it the connection wasn’t superficial either. It wasn’t just, there’s a guy calling himself John Titor from the future on 2ch. A lot of details, like the IBM 5100 and the mechanics of time travel and divergence numbers, are taken straight from the story. You can think of Steins;Gate as one giant John Titor reference.

This is fascinating to me, because even though I’m not intimately familiar with the details of the John Titor story, I’ve been on the Internet long enough to remember this whole thing. It’s kind of weird to see something that was sort of the early-web version of an internet meme be used as the skeleton of a more fleshed-out story. I guess something like Densha Otoko would qualify as well, but I think there’s a lot less you can do with it to make something unique like Steins;Gate.

This whole aspect of Steins;Gate got me because I am a nerd and nerds friggin love references. I mean, look at any Shaft anime. But yes, Steins;Gate has tons of references and where there aren’t any, it’ll just create memes. For the most part, these references are all internet memes. If it’s not part of the John Titor mythos, they’ll be working in a 2ch meme. And then there’s stuff like EL PSY CONGROO and tutturu~ which seems like it was designed to be spammed across the internet.

The characters themselves are exactly the same kinds of people. After all, they encounter John Titor on 2ch. What kind of people read and post on 2ch? Daru and Okarin are really obvious otaku. But Mayuri seems relatively normal, except that she works at a maid cafe in Akiba and buys doujin at Toranoana. Okay. Even the most respectable and well-adjusted of the cast, Kurisu-TINA, is a closet VIPPER.

So I guess what’s special about Steins;Gate isn’t that it’s just a good story about a bunch of friends who fall into some time travel conspiracy. It’s that it’s a good story about a bunch of friends who are just like the people watching it who fall into a time travel conspiracy based on an internet urban legend.

「一面の花」/「k野」

I’d been hearing about how awesome Heartcatch Precure was so I’d always intended to eventually watch it. The art was fantastic and I’d seen some great clips of the fights. The problem is that when I found out about it, it was already really far into the 50 or so episodes. So once I got a ton of time after I graduated and was looking for a show to watch, I figured it was time. After all, I rolled through Cardcaptor Sakura pretty quickly.

I’ve been thinking about whether I’d put Heartcatch at the top of my magical girl list, but I think it ends up coming second to Cardcaptor Sakura. Still, this means I think that Heartcatch is pretty damned good. I think it’s a better magical girl show than Madoka or Nanoha and I friggin love those two shows.

What makes Heartcatch so great is that it’s just good. It has a great cast of characters, the story is wonderful, and the production is stellar. Most importantly, it’s just a magical girl show. It isn’t magical girl with lasers. It isn’t magical girl with zombies. It is what it is and it stands toe to toe with all of those magical girl and x shows, where x is something that makes it more palatable to people who need an excuse for why they’re watching shows for little girls.

And Heartcatch is girly as hell. Yeah, the Cures might be throwing some punches around, but the show is tripping flowers and fashion. You can attempt to reframe Nanoha as scifi and talk about intelligent devices and stuff, but the Cures spray perfume to transform. At this point, Nanoha’s pretty much decked out in mecha gear in Force. Meanwhile, this is a show with tambourines and those spinny things that look way too much like their only function is to sell toys.

There’s also the characters’ boundless optimism and energy. There isn’t anything that’s especially dark about is show, even the villains are kind of goofy. Everyone’s got dreams and everyone’s fighting to protect their dreams and it’s terrible that villains use the uncertainty in people’s hearts against them. Here, the objective isn’t to seal away some rogue magics or befriend the opposition with lasers, but to restore people’s hearts.

The other Precure series are going to have a tough time measuring up to the Heartcatch bar, but I hope they do.

「ガンバスター」/「タガナー」

It’s quite illuminating to watch Gunbuster and recognize things that I’ve seen before scattered throughout so many shows just because they decided to pay an homage to this. The obvious case is Gurren Lagann, where certain little things are straight up ripped out from this show. Another case would be Lucky Star’s Bun Guster track. At many of these points, it was like a light flipped on in my head and I went ohhhhhhhhhhhh.

What’s more impressive is that even though this is one of the oldest shows I’ve watched, it still felt new. It was kind of like when I watched Utena for the first time. In this case, the time dilation stuff added a really interesting dimension to the entire plot. While Gurren Lagann relies on physical size alone to give it that grand sense of scale, here, time and distance are used as well. It’s not on the same scale physically, but it definitely matches in the sort of awe you feel when you watch it.

This all makes Diebuster quite interesting as well. While it’s a “sequel”, it feels very different. There are some callbacks and some of the same themes, but it’s largely doing its own thing. I spent a lot of time trying to figure it out and how it connected to Gunbuster but it just ended up giving me all the answers I was looking for. I don’t think there was the same sort of awe as when I watched Gunbuster, but it was more like ‘wow, that’s pretty neat’.

But comparing these two to Gurren Lagann makes their similarities a lot more apparent. Both are about a girl who look up to a girl who’s very successful as their role model. Both of them eventually surpass their role model and achieve their dreams through hard work and guts. And then both of them save the world together with the mentor they’ve surpassed.

Watching these was kind of exciting because everyone loves Gurren Lagann and the question is whether Gainax will be able to top it. I imagine everyone who watched Gunbuster and Diebuster must’ve thought the same thing, that surely they can’t top THIS. I’m glad that they’ve proven us wrong twice now.

「静雄無双」/「巖本英利/旧PNス・タンリー」

I’d planned to write about Durarara!! during the whole twelve days of anime thing, but I dropped it at the last minute to make room for Utena and Star Driver. But now, you’re going, wait a second, there was no Durarararararara!! this year, how can it be a special moment in anime for this year?

Well, you see, it turns out a chunk of the light novels was decently far along. I went through books four through six, which, if we allow ourselves to dream a bit, would correspond to the second season. I liked the anime, but I think everyone is in agreement that the second half is weaker, although whether it was still good or terrible is up for debate. The nice thing about the three books immediately after is that the focus is not on Anri or Kida, so you won’t have to worry about them being a downer in your anime about crazy and exciting things in the big city.

So it should please everyone that a second season of the anime focuses much, much more heavily on everyone’s favourite really angry guy, Heiwajima Shizuo. The fourth book goes back to the Durarara/Baccano standard of telling a story with about a thousand different threads somehow coming together at the end. The fifth and sixth books form one story in which Shizuo is the focus in the same way that Mikado, Anri, and Kida were the main characters of their respective arcs. We’ll also see some familiar faces. An interesting thing about the fourth book is the return of some characters that you may or may not remember and the introduction of several new ones.

Shizuo was a really great side character before. He shows up, gets trolled by Izaya, and proceeds to destroy things, usually with other things. The focus that books 5 and 6 puts on him is great because it develops him from the guy that can be described by the previous sentence into a legitimate badass. I feel like he’s always been thoughtful and that his only real problem is getting pissed off really easily, so I enjoyed watching him put his talents to use in a more focused way.

The thing that I enjoy about Ryohgo Narita’s stuff is his ability to launch a million seemingly unrelated characters and plot threads and be able to tie them together. The one-volume version of that is very neat to read through, but I have a feeling that there’s some longer term stuff that he’s set up in the early books that will come back in a big way and there are already hints of that at the end of the sixth book. This isn’t unlike the Haruhi series, where seemingly inconsequential details come back later in an earthshattering way, although, here the pieces are bigger, I guess.

Really looking forward to that second season.

「ありがとう放浪息子」/「神田川」

The main problem for the characters is some variation of that, isn’t it?

Anime is full of traps and when you sit back and think about it, it’s kind of like, wow, they’re everywhere, aren’t they? Which, if you think about it, is kind of strange that Hourou Musuko feels so different from everything else. I mean, it’s an anime about traps.

An easy description of Hourou Musuko would be that it’s about gender and sexual identity and it does it realistically, unlike the usual fetish or comedy treatment. But what’s more interesting isn’t just that it examines these things in a realistic and mature manner, but that it starts from square one. These are children that are dealing with these things.

So we have these kids and we’re watching them struggle with it on their own. If they’re lucky, they meet someone to struggle along with. Maybe they’re doing okay for a while. Suddenly, things change, and they’re in middle school and have a whole new slew of bullshit to worry about. There’s never really any rest from these things, there are no answers and the kids just have to keep on growing up and dealing with crap that comes their way.

That the anime chose to focus on the middle school parts was something I found kind of unfortunate. Yeah, it’s really exciting that there’s a ton of relationship drama here, but I thought a nice thing about Hourou Musuko was that it took the time to take a look at the slower parts of the characters’ lives as well. Of course, it worked out that the particular slice that they chose to animate has some sort of closure, so that was nice. But just like how the manga starts long before the anime does, now the manga has gone beyond where the anime ends and everyone’s in high school.

Just like with Usagi Drop, we’ve had our fun times watching these kids grow up, but that’s not going to last forever. I don’t know whether this mangaka typically focuses on long term things, but I’d really like to see where this crew ends up in five, ten, or more years down the road.

「闇の力を秘めし鍵よ」/「hatsuko」

I don’t think it’s that much of a surprise if I tell you that Madoka made me watch Cardcaptor Sakura. Well, that’s not entirely true. A more accurate phrasing of that statement would be, I watched Cardcaptor Sakura in anticipation of Madoka. Yep, I spent a good chunk of the Christmas break just before Madoka aired watching the entirety of Cardcaptor Sakura.

Cardcaptor Sakura seems to be one of those childhood anime that sits alongside the likes of Dragon Ball or Sailor Moon. I actually had no idea it was dubbed in English and aired here until when I was way too old and cool for that crap. More recently, it went from ‘that show that a few of my friends watched that seemed lame’ to ‘that show that people cite as the epitome of its genre’.

CCS is a pretty lengthy show, at about 70 episodes. After I got into the whole magical girl thing, I’d planned to watch it eventually, but never really had the time or motivation to do so (which is fairly common for shows that I plan to watch because I “should”). But with 2011 and Madoka quickly approaching and having finished all of my grad school applications, I convinced myself to finally start.

Up until then, my benchmark for magical girl shows was Nanoha. As far as Sakura and Nanoha go, they’re pretty similar. For a while, before I watched either of them, I’d get them confused sometimes. Obviously, I knew that CCS wasn’t going to have huge explosions or funnels, but it was sort of surprising because I’d forgotten that magic doesn’t necessarily manifest itself as a giant pink beam of death. It turns out magic has a wide array of uses.

Speaking of magic, I was surprised to learn that Sakura’s magical girl outfit was not and that there wasn’t even a default outfit. I don’t even know if she actually ever wears the one that she’s seen wearing in most promotional art in the show. The only constant in her magical girl wardrobe is her staff. But nope, Sakura’s eclectic fashion is thanks to Tomoyo.

And I’m not entirely sure when it happened but the focus of the show seemed to shift slightly from Sakura learning to be a magical girl to include falling in love. She basically spends the entire show tracking down the Clow Cards in some way, but somewhere along the line we also get scenes of her and Syaoran sorting their feelings out. Something that I didn’t expect was for the entire show to basically consist of Sakura dealing with the Clow Cards. But even then, the last thing that Sakura had to do before everything ended wasn’t to save the world, it was to tell Syaoran that she loved him.

This love thing was new to me, especially coming from Nanoha, where at best, we have some best friends forever stuff crossing the line occasionally with undertones. Girls and boys interested in each other? Why I never! But it was interesting because it actually went there instead of dancing around it. And it was quite nice seeing Sakura and Syaoran growing in that way.

I think this show made me realize it’s okay to watch shows made for little girls even if they’re not wrapped in a layer of beam weaponry and sci-fi military trappings.

P.S. despite this post’s title, the best OP is Platinum

「キャスターと龍之介」/「ラク」

It’s that time of year again, in which I brush off a bunch of drafts that I’d hoped to post over the course of the year but didn’t get around to finishing on time. A funny thing happened on the way here, which is that I was expecting to not have enough to post about. The reason is that I would have been posting consistently about all the things I wanted to post about all this time. But as it turns out, well, “@blkmage ha ha haaaaa”.

Look forward to twelve days of quality handcrafted posts about some japtoons.

An introduction to mathematical writing

In chapter 2 of Suugaku Girl, we’re introduced to the third component of the little love triangle that’s forming. Tetra is the underclassman that the main character is tutoring and she’s one of the many people who think they might like math but school eventually beats that silly thought out of them.

Anyone who’s taken a real math class will know that by the end of it, your notes are essentially a giant list of definitions, theorems, and proofs along with an example thrown in once in a while. By a real math class, I mean a math class that’s actually concerned about reasoning about how we get our theorems instead of focusing on how to use them for practical applications. I did my undergrad at a school where math has its own faculty and where there’s a lot of rivalry between the various faculties. Everyone in math often jokes about how an example suffices as a proof for engineers and their ilk.

I checked your textbook for a proof and it said that we’ve done enough examples for it to be plausible. Must have been written by engineers.
— Vanderburg, PMATH 340 (F09)

Here, Tetra falls into the same trap when the main character asks her to define what prime numbers are. So MC-kun has to explain how definitions work. First of all, definitions have to be precise, just like theorems. It’s pretty easy to lose marks for misstating theorems and definitions by leaving off an edge case for 0 or 1 or something silly like that. Of course, that leads Tetra to question why these things need to be so precise and arbitrary.

One of the things I realized about math is that it’s all about trying to do things with the definitions you start out with. You can kind of see that in how the number system is built up. We can start with plain old numbers that we use to count things. And then we can add things to it. Like, what happens if we have less of a thing, how would we represent that? Tada, we’ve got negative numbers. Okay, now what if we have a part of a thing? Now we’ve got fractions. And so on and so forth. We realize that not every number can be written as a fraction and suddenly we’ve defined real numbers.

I’m sure we’ve all wondered why complex numbers work out the way they do. It’s because we’ve defined everything to work out like that. Some guy tried to take the square root of a negative number and found that it didn’t work out very well, so we defined the square root of -1 to be $i$. Now we have this $i$ thing, what can we do with it? Well, we can just start writing crazy things like $3 + i + 4i^2 + 5i^3 + 9i^4 + 2i^5 + 6i^6$, but then we realize that it just becomes $2-2i$. Well, okay that’s kind of neat.

But now we know that all complex numbers can be written as $a+bi$ with $a,b\in\mathbb{R}$. So someone along the line must’ve been like, what would happen if we tried to graph these things? So we treat $a$ as the $x$ component and $b$ as the $y$ component and it turns out we can think of complex numbers as other structures like a vector or just a 2-tuple or something. And suddenly, this gives us a way to compare complex numbers, by taking the length of the vector that they define. And now that we have vectors, we can do some weird geometry stuff with them. We can think of these things as the length of the vector and the angle that they form. And then you can go crazy and talk about roots of unity or what multiplication of complex numbers might mean.

The point is that all of mathematics is built up like this. You start off with some definitions or premises and you go nuts. Of course, you don’t have to use the same definitions as everyone else. The problem with doing that is you might not end up with anything interesting. Like, what if we did something nuts and defined 1 to be a prime number like so many people do when they leave off that condition? Well, not much, we’d probably just rephrase all of our theorems to exclude 1.

On the other hand, sometimes when we play around with definitions, we do end up with something interesting. For instance, we can define something called the extended complex numbers, which is just the set $\mathbb{C}\cup\{\infty\}$. Yep, we just say okay, infinity is a number now, deal with it. So what can we do now?

Well, we can divide by 0 now.

I imagine there might be a few people who might flip out at this notion, but yes, since $\infty$ is included in our number system, we can define $\frac{1}{0}=\infty$. Of course, we can’t do everything — $0\times\infty$ and $\infty-\infty$ still don’t mean anything. But if you’ve been paying attention, you might be going, okay well, we can divide by 0, but what else can we do? Dividing by 0 is kinda meaningless if there’s nothing new we can do.

As it turns out, the extended complex numbers defines a very different geometric object. If we remember from the example of the complex numbers, we can basically define every complex number as a vector over a two-dimensional plane. Here, with the extended complex numbers, we can define every number as a line passing through something called the Riemann sphere. And like the complex plane, this sphere lets us create weird relationships between numbers and angles and stuff. This turns out to have interesting properties in complex analysis and quantum mechanics.

So yes, definitions are often arbitrary. Why? Because it’s just useful and interesting that way. You could argue that it’s because nature forces us to define things a certain way. Kind of like, of course you can’t take a square root of a negative number, you just can’t do it! What happens, though, is that we always seem to end up finding useful things that line up with our mathematics rather than inventing our mathematics to do useful things with. After all, mathematicians have been playing around with imaginary numbers for at least a few decades before electromagnetism was even discovered.