# The 2nd annual π day anime and mathematics post

「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.

# 12 Days V:

「消失」/「Me」

Much to everyone’s surprise, Haruhi reappeared suddenly in the summer of 2009. Much to everyone’s surprise, one particular chapter was extended eight times its length and earned the ire of pretty much everyone. Much to everyone’s surprise, the thing that everyone thought they’d finally see would finally happen, even if it was half a year later in a movie theatre.

To be honest, I think the thing I am most mad about regarding Endless Eight is that it pretty much poisoned the Haruhi well and has made it impossible to convince some people to be interested in Disappearance. And that’s terrible, because Disappearance is the best part of the entire series.

Even having read Disappearance about a year before and even watching the awful eyecancer version of the movie, it was still fantastic. And I imagine I’ll be glued to it when I watch it again in glorious 1080p.

The thing about Disappearance is that it’s so different from whatever came before it. I mean, an obvious cause is Haruhi’s absence, what with her being the pivot that the other characters hover around. But it really feels like it’s in a different genre. Beyond the first arc of the anime, there really aren’t any other huge moments like the ones that are delivered in the movie. And I’d say that’s because, beyond the first arc, there wasn’t really any reason to until now.

And so, unlike most of the short stories we’ve seen, Disappearance takes the hugest leaps in terms of character development and the general direction of the entire series since it started. I mean, there’s a reason why the only volumes of the light novel that are sitting on my shelf right now are Melancholy and Disappearance. This is really the first indication that there’s something more to this series than random SOS-dan shenanigans.

The movie isn’t quite as fun as you might expect, but it is pretty enthralling like a good mystery should be. But most importantly, it assured me that my tastes weren’t entirely questionable. I was right the first time: Haruhi was something special.

# The （　´_ゝ`） of Haruhi Suzumiya

So the Sighs of Haruhi Suzumiya follows ENDLESS EIGHT ENDLESS EIGHT ENDLESS EIGHT ENDLESS EIGHT. With that, the second season of Haruhi is cemented in mediocrity.

Reading the second volume of the light novel after having seen short stories from the third and fifth volumes was pretty interesting. There are a few gems in there, but in general, it’s not quite as strong as some of the other volumes. That’s understandable, though. It was only the second volume and I can see how cool it would have been to see the resultant movie several volumes later. It’s expected that the later volumes would have stronger writing.

Still, Mikuru shooting stuff out of her eyes and Yuki saving the day is great. Shamisen and his philosophical insights are also great. Kyon getting pushed to the limits of his patience is fantastic. But why isn’t Sighs enough to placate the rage over ENDLESS EIGHT ENDLESS EIGHT ENDLESS EIGHT? Isn’t it not ENDLESS EIGHT ENDLESS EIGHT ENDLESS EIGHT which means it’s great?

Again, let’s consider what was animated this season: Bamboo Leaf Rhapsody, Endless Eight, and Sighs. What did the predicted broadcast indicate? Bamboo Leaf Rhapsody, Endless Eight, Sighs, Disappearance, and one or two other chapters. Instead of animating arguably the best arc of the series and another chapter or two, we got the same chapter eight times.

It doesn’t matter whether or not Sighs is a return to form and that it’s not ENDLESS EIGHT ENDLESS EIGHT ENDLESS EIGHT ENDLESS EIGHT. Sighs isn’t Disappearance. Instead of seeing crazy time travel and alternate universes, we got the making of The Adventures of Mikuru Asahina Episode 00. Instead of seeing an arc with an enormous amount of character development, we got more of the same. Instead of getting an epic capstone arc, we got more supplementary chapters.

The thing that is most disappointing is that they went out of their way to deliver a mediocre season, when all they had to do was do a straight adaptation. I guess I’ll be looking forward to Little Busters! instead of the next Haruhi, because at least they don’t screw around with Key games.

# Endless Eight: Endless Eight endless eight endless eight

### Endless Eight

So, Endless Eight. I’m not sure what it is about Haruhi that makes every party involved go insane, but we’ve seen it happen to fans, the author, the publisher, and now the studio. I’m probably going to be giving Kyoto Animation the benefit of the doubt and say that they are inept and not malicious, because honestly, they’d be inept either way. From that interview about how they went about producing the arc, it sounds like they genuinely thought that they were being brilliant artistes.

There’s a theory that Kyoto Animation did it to manage expectations. That theory is dumb. All they had to do was deliver a solid and close adaptation of the light novels into anime form. This isn’t some monumentally impossible task like it is for Umineko. The theory also makes no sense. It’s like saying the next light novel volume needs to be tripe in order to make the following one more successful.

### Endless Eight

The finale of Endless Eight shows us that it was possible to do the entire chapter in one episode, much like Bamboo Leaf Rhapsody. I can’t believe it either, but some people liked watching seven more episodes of the SOS-dan doing mundane things. I wonder if they would have still liked it if they understood the squandered opportunities that seven episodes would have delivered:

• Sighs
• Where did the Cat Go? and Snow Mountain Syndrome
• Charmed at First Sight Lover
• The Melancholy of Mikuru Asahina
• Editor in Chief Striaght Ahead!

All of these have either more interesting stuff that the SOS-dan does or contains more character development. Managing expectations? My expectation was for another season of solid side stories capped off with Disappearance. What I got was two side stories, a waste of half a season, and the lingering fear that the studio still has some trolls up its sleeve. And we haven’t even gotten to Disappearance yet.

So, Endless Eight. I’m not sure what it is about Haruhi that makes every party involved go insane, but we’ve seen it happen to fans, the author, the publisher, and now the studio. I’m probably going to be giving Kyoto Animation the benefit of the doubt and say that they are inept and not malicious, because honestly, they’d be inept either way. From that interview about how they went about producing the arc, it sounds like they genuinely thought that they were being brilliant artistes.

### Endless Eight

There’s a theory that Kyoto Animation did it to manage expectations. That theory is dumb. All they had to do was deliver a solid and close adaptation of the light novels into anime form. This isn’t some monumentally impossible task like it is for Umineko. The theory also makes no sense. It’s like saying the next light novel volume needs to be tripe in order to make the following one more successful.

The finale of Endless Eight shows us that it was possible to do the entire chapter in one episode, much like Bamboo Leaf Rhapsody. I can’t believe it either, but some people liked watching seven more episodes of the SOS-dan doing mundane things. I wonder if they would have still liked it if they understood the squandered opportunities that seven episodes would have delivered:

• Sighs
• Where did the Cat Go? and Snow Mountain Syndrome
• Charmed at First Sight Lover
• The Melancholy of Mikuru Asahina
• Editor in Chief Striaght Ahead!

### Endless Eight

All of these have either more interesting stuff that the SOS-dan does or contains more character development. Managing expectations? My expectation was for another season of solid side stories capped off with Disappearance. What I got was two side stories, a waste of half a season, and the lingering fear that the studio still has some trolls up its sleeve. And we haven’t even gotten to Disappearance yet.

# Light novel talk: Suzumiya Haruhi

It hadn’t occured to me until a week before the new episode of Haruhi aired that I could have just read the light novel translations on Baka-Tsuki instead of waiting three years for the new anime. In fact that us usually what I do for most other anime: watch as much as is available and go to other sources, usually the manga.

The problem with light novels is that they’re a lot harder to translate. For anime and manga, there’s really only dialogue and whatever signs are on screen. Light novels however, are entirely text. In anime and manga, if you’re a bad translator, you have visual cues to help the viewer understand your crappy translating. For light novels, the reader’s understanding rests entirely in the text, and therefore, your translation.

As a result, light novels take a much longer time to translate. They’re usually far behind their anime adaptations, usually because they only get popular after they get an anime. It’s only in instances like Haruhi where the publisher screws around with the audience for three years that we get complete translations of the volumes that are available ahead of the anime.

The light novel volumes are arranged so that they’re either a series of short stories or they contain one plot arc. For instance, the anime animated one volume that contained a plot arc, Melancholy, while all the others were short stories from other volumes. For the most part, the important things happen in the arc volumes with some incidental stuff happening in the shorts.

The most interesting thing about the light novels is the focus on Haruhi is lessened and we get to see the other characters grow and do stuff. There are entire plot arcs where Haruhi is not the focus. Sure, she’s the focus and reason of the SOS-dan and weird stuff still happens because of her, but she’s not always the one being examined and she’s not always the one who has problems.

It becomes clear Haruhi isn’t just a bunch of random stories about the SOS-dan. Stringing the plot arcs together, you can see a narrative being woven. Some things that make this clear are the conflicts that come up. The anime doesn’t make it too clear that the Organization, Time Travellers, and Data Entity aren’t necessarily allied and that there are tensions between Yuki, Mikuru, and Itsuki at first. Eventually, they become friends through the SOS-dan, but it’s not until the appearance of opposing factions that their superiors officially start working together.

Each character has their own struggles. Yuki, created to interface with humans, wants to become more human. Mikiru, who has information withheld from her by her superiors and relies on Kyon, wants to become less useless. Itsuki, who used to be normal until he got his esper powers, begins to have trouble keeping up his facade and wants to be normal again.

The later volumes of the light novels are especially fun. I mentioned that the short stories contain incidental information. This doesn’t mean they’re pointless. They contain some character development. More importantly, they contain obscure details that the author sets up to use in later, more important plot arcs. I really enjoy that because random details come back to fix problems later on. The author doesn’t forget what happened or what he’s set up from earlier.

Is it worth reading the novels now that the second season is reality? Yes, especially since the last few volumes won’t be animated, and it’s these that contain a lot of the meta-conflict. But, if you’re enjoying the anime, you might want to wait until it’s over before you dive in so you don’t spoil yourself. because lol Endless Eight.