12 Days VIII: Actually, I don’t think I quite caught the bunny allusion

「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.

12 Days II: My body is lady

「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.

Suugaku Girl supplementary handout for chapter 2: On definitions

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.

Suugaku Girl supplementary handout for chapter 1: Sequences

That really isn't enough terms to identify as Fibonacci

Suugaku Girl is a manga about people who really like math. The great thing about it is that it contains a substantial amount of math. This is also great because it will give me plenty to blog about.

The first chapter of Suugaku Girl is about the main character’s initial meeting with Milka, our socially retarded genius meganekko. She just starts spouting out numbers and, for whatever reason, he feels compelled to guess what comes next. And that is apparently the beginning of this love story.

In the chapter, we’re introduced to a few sequences, some of which are famous and some of which you might not have considered a sequence. It’ll probably help to understand what a sequence is beyond just being a bunch of numbers in a prescribed order.

Formally, we define a sequence as a function $a:S\to R$. The set $S$ is basically our index and is $\{1,2,…,n\}$ if $a$ is a finite sequence or the set of natural numbers $\mathbb{N}$ if we’re dealing with an infinite sequence. Normally, functions are written as $a(n)$, but, as was alluded to before, we refer to terms by index as in $a_n$.

However, $R$ can be any set. In the case of Suugaku Girl, it seems to be sticking with integers, but we can have sequences of bitstrings, vectors, complex numbers, functions, or whatever. What this also means is that a sequence doesn’t necessarily need to have a “pattern”, but can really be any ordered list of numbers (or functions or vectors or…).

Milka also brings up the idea of infinite sequences. A lot of the time, people will try to “solve” a sequence by completing it when they’re given only the first few terms. But, like Milka suggests, what they’re doing in that case is assuming that the rest of the sequence goes on as initially implied. Really, we can define any sequence we like with any behaviour we like. Again, remember that a sequence can be anything we want it to be. In fact, the sequence that Milka defines using the digits of π is kind of like that in that it’s completely arbitrary and doesn’t really have the kind of sequence definitions we’re used to seeing.

For finite sequences, we can just list all of the terms of our sequence like $(1,1,1,1,1,1000000,1)$. Obviously, we can’t do that for an infinite sequence. Luckily, a lot of the time we define sequences that have some sort of useful pattern that we can represent in a succinct way. Sometimes, like the digits of π sequence, this is harder.

We can try to formally define all of the sequences that were given in the manga. For instance, the Fibonacci sequence $(a_n)_{n=1}^\infty$ is commonly defined as $a_n = a_{n-2} + a_{n-1}$, but we have to give the first two terms $a_1 = 1, a_2 = 1$. The second sequence (which we’ll call $(b_n)_{n=1}^\infty$) takes a bit more work to define. We’ll need to define $p_n$ to be the $n$th prime number and then we can define $b_n = p_n \times p_{n+1}$. The third sequence $(c_n)_{n=1}^\infty$ is easy, it’s just $c_n = n^n$. And we can formalize the last one, which I’ll call $(d_n)_{n=1}^\infty$, just like the first two with a few more words. We let $\pi = q_1q_2q_3\cdots$ be the decimal expansion of π. Then $d_n = 2\cdot q_n$.

So that’s all fine, but what exactly are sequences used for? I’m pretty sure everyone’s learned about arithmetic and geometric sequences in grade school. Obviously, we can study sequences and their behaviour on their own. We can talk about whether they increase or decrease or how fast they grow or whether they converge. Apart from that, I don’t remember seeing sequences used for something besides sequences until analysis.

Analysis is basically the field of pure math that formalizes the concepts that we’re introduced to in calculus and generalizes them to spaces. Limits are a fundamental idea in calculus and analysis and these are defined by how a sequence converges. And this is where those weird sequences of vectors or functions comes into play, since we can talk about the convergence of a sequence of vectors or a sequence of functions in these other spaces that we want to do analysis in.

That’s probably the easiest example of an “application” of sequences. For myself, over the last few months I’ve read about automatic sequences, which are sequences that can be generated by a deterministic finite automaton. This gives us a way to relate automata theory to number theory and algebra. For instance, once we have k-automatic sequences, we can talk about k-regular sequences and come up with power series with respect to certain rings and fields and bla bla bla.

If you ever want to find out what crazy sequence you might have a part of, check out the Online Encyclopedia of Integer Sequences.

12 Days XII: Life sucks

「punpun」/「m2」

「punpun」/「m2」

That’s what Asano Inio is trying to say. Well, okay, there’s a bit more to that: life sucks, but, you know, It’s going to be okay.

I’ve already talked about Solanin, but since then, I’ve been trying to read everything I can that’s by this guy. The bulk of it was various one-shots and short stories that weren’t more than a handful of chapters. Even though they’re short, he’s able to connect you with the characters and the hopelessness or boredom of their situation. Nijigahara Holograph is really his one step away from talking about peoples’ lives. It’s about some supernatural happenings, which when combined with his ability to write dark and realistic situations, make for a really creepy story.

But the one that stands out to me besides Solanin is also one of his strangest, Oyasumi Punpun.

Oyasumi Punpun is about a kid, Punpun, who is just a normal boy. It’s about him growing up. We start with him in elementary school. We see him play around with his friends. We see him wrestle with his feelings as he realizes he has his first crush on a girl. We see him deal with his family issues, with his abusive father and alcoholic mother. We see him move on to various stages of his life.

Except that to the reader, he and his entire immediate family are rendered as really weird, flat bird creatures.

Trust me, it works very well.

Oyasumi Punpun starts off sort of whimsically, with the crazy God character popping up once in a while and the surreal imagery that comes up in Punpun’s thoughts. Obviously, the family situation makes it pretty dark as well, but later on it gets fairly depressing. It’s a lot like Solanin, except that Solanin focused on a very, very short snapshot of time in Meiko’s life. In Punpun, we’re watching Punpun grow up and we can see how all of the stuff that he encounters earlier on in his life goes on to affect him later on.

What’s ultimately depressing about Punpun isn’t that it’s Punpun going through all of the crap that he goes through. It’s realizing that the things that he goes through are entirely believable, that real people go through what he has to go through.

And that’s what’s really so amazing about Asano Inio. It doesn’t matter whether or not the character is like you, which was the case for me with Solanin. Heck, it doesn’t even matter whether the character looks anything remotely close to a person. He’s still able to make what they’re going through and what they’re feeling uncomfortably real.