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It starts, as these things always do, with graffiti.

Neku’s still growing into his new role, takes to wandering. Apparently Kitianiji had as well, before his Entry Fee had robbed him of the right to leave the Dead God’s Pad -- and though Neku spent weeks trying to forget the Long Game, tried to purge it from his mind the way it had been purged from the collective mind of the entire world, memories have a tendency to linger like a bad odor.

So he rests his headphones on his shoulders and wanders like Kitaniji had wandered, up and down alleyways, to rooftops and basement clubs, fiddles with his frequency so when he’s nearby the living get a rush of unexplained creative euphoria and the dead get the chance to have a conversation with a stranger. That’s something that Kitaniji, with his undying, almost fanatical devotion to the Composer and the rules of the Game, had never done; and even if it means that every now and then there’s a kid on Day 7 going oh shit, you’re that guy from the thing in Spain Hill, or holy crap, I think it’s the dude from the store on Cat Street -- well, that’s just how it is.

Not that Neku’s going to spend the rest of eternity comparing himself to Kitaniji, no siree. He’s having enough fun playing psychopomp-slash-lapdog to a sociopathic boy-king.

It’s all entirely unrelated to how much of Kitaniji persisted, in the music he’d produced and the trends he’d set and the look in the eyes of certain people who were old enough to have been around before Neku’s tenure, and Neku figured he needed at least five pins to collect all that stray Soul. He puts in the request for the blanks to Joshua, who says he’ll pass it on to the Producer, and then the Composer turns into himself and mopes for ten straight days during which a fine grey mist rains down on Shibuya and everyone feels generally like shit.

It’s supposed to be a secret who the Producer was, but nothing about the Long Game had gone according to anyone’s plans but his -- and he was the kind of guy who you’d just know was someone special after talking to him, like, once.

Neku had thought it was the CAT thing, but CAT was only half of it. And -- well, Neku is CAT now, and when the five blank pins arrive it’s clear that Shibuya has a new Producer, and despite everything you can’t help the fact that life and death and life go on.

So Neku wanders. He takes the blanks with him and gathers stray bits of Kitaniji, starts weaving them into a coherent story, enjoys Shibuya for what it is and what it is constantly becoming. He spends days sitting on a balcony overlooking the Scramble, the cacophony of real life only slightly out of tune with the noise of the Game; he ambles through Miyashita Park; he shares a cup of coffee with a pair of bewildered Players; he watches some junior Reapers crunch a pair of Players and then wastes an afternoon watching a river of static that had once been human souls wind its way through Molco towards the Shibuya River.

Then there’s an afternoon in the late spring when he wanders into Udagawa, and someone has tagged over his work.

That’s normal -- hell if Neku’s the only graffiti artist in Shibuya, and he’s not even the only Reaper who likes to screw around with paint. What’s different is the way he feels when he looks at the tag, something buzzing in his brain like a shot of espresso or a bolt from the blue, and he knows that it’s not your run-of-the-mill kid with a spray can. It’s a simple image, of a repeating wave split down the middle with a horizontal line -- it reminds him of something he can neither place or name. It disturbs him, though he has no idea why. What the hell is it? He’s seen it before -- in a textbook maybe -- back at school?

(Downside to dying at sixteen is you never finish high school. Upside to dying at sixteen is you don’t have to finish high school.)

Neku wanders back to his old high school in the middle of the night, flits from classroom to classroom, avoids looking up names of current or graduated students, half-heartedly flips through textbooks looking for an image of a wave laid on top of a Cartesian plane --

Oh, fuck me, he thinks, dropping the chemistry text, making a beeline for the math department office. Somebody talented at manipulating Soul has tagged his mural with a well-labelled graph of a cosine function, and there is one and only one motherfucker in the universe both interested in and capable of doing something like that, and he’s supposed to be dead.

 


 

The set of natural numbers (that is, the nonzero positive whole numbers) nests inside the set of integers (that is, whole numbers both positive and negative, an infinitely long line with zero in its centre), which nests inside the set of rational numbers (that is, numbers that can be expressed as the fraction integer over integer, often represented as decimals that either terminate or repeat). The set of real numbers can then be split into the rational and the irrational numbers, with complex and nonexistent numbers on its periphery, the imaginary unit i floating about the Matryoshka doll of real number like a ghost haunting the house of the soul.

In the late nineteenth century Georg Cantor had showed, through a proof so simple one might call it profound, that the size of the set of natural numbers is the same as the size of the set of integers and the same as the size of the set of rational numbers -- that despite what intuition tells us, that despite the fact that it would seem that there are twice as many whole numbers as even numbers, the infinities of countable sets are all the same size.

How strange that these sets that nest inside each other are the same size, but that’s what happens when your eggs are infinitely large, the size of a dream or a thought or a city that only exists inside one’s imagination, and there’s no accounting for mathematics, and certainly even less accounting for mathematicians.

 


  

“Megumi showed me more respect,” Shibuya whines, because even after everything Shibuya is a petulant child. “He’d have bowed and scraped a little more. Made me dinner. Asked for my forgiveness, even if he’d done nothing wrong.”

Neku says nothing about Kitaniji. Time has taught him that this is Joshua’s complicated way of processing guilt -- gods are not built for that category of emotion. Then: “You killed the Grim Heaper, didn’t you?”

“Sure I did. He was annoying me with all his grandstanding about Konishi this and zetta that. Why do you ask?”

There’s an almost imperceptible shift in the laws of gravity, and the key of A minor shifts up a semi-semi-semitone. Joshua’s hiding something.

“Someone tagged the mural in Udagawa with a sinusoidal function.”

“Why, Neku. I had no idea you knew so many big words.” Definitely hiding something. People in Shibuya have a tendency to change the subject when the conversation gets dangerous.

“Whatever,” Neku says, instead of anything else, because the unspoken trust between them -- that Joshua will not imprint on his Conductor to get him to drop sensitive subjects, and that Neku will not betray the confidence of a Composer he’s comfortable arguing with -- is far more important than whether or not someone has come back from the dead.

 


 

The cosine function is a periodic sinusoid whose critical points are derived from the unit circle: a three hundred sixty degree rotation counterclockwise from the point (1,0), henceforth referred to as 2𝝅 radians, traces a circle which Occam’s Razor and Euclid’s soft geometry of similar triangles allows us to assume has a radius of one. Any point on the unit circle therefore has cartesian coordinates (x, y) = (cosθ, sinθ) -- this is derived from how the opposite and adjacent sides of a right triangle relate to the hypotenuse that is the terminal arm -- where θ is the angle in standard position. The cosine sinusoid hits its zeroes at the coterminal angles of 𝝅/2 and 3𝝅/2, as shown:



 

Of course, that’s all quite useless to someone who doesn’t need to derive Euler’s Identity or take over Shibuya any time soon. This is the kind of mathematics whose very existence is a philosophical puzzle, whose primary purpose is nothing but the beauty of abstract patternmaking -- and we haven’t even gotten into trig identities, the hours Minamimoto spent as a fifth-grader puzzling over left-side/right-side tables, the ways that an expression could be transformed into something a more palatable with a little elbow grease and a decent understanding of rational equivalencies, and how an equals sign will gleefully betray you the instant you abandon equations for proofs.

Where’s your beauty?

 


 

 

How cruel of the narrative to take anyone he could talk to from him at the very moment he was learning to open up his world. Neku would give his left wing for five minutes with Mr. H, or ten minutes where Joshua could only tell straight-forward truths, or a genuine friendship with Lollipop unburdened by the past. The latter seems the most likely, so he spends a Game quietly contemplating the cosine wave in Udagawa, then on Day 7 meanders to Dogenzaka where the Game Masters are mowing down players before their traditional celebratory ramen lunch.

“Good Game,” Neku says, over the tonkotsu broth Ken Doi always foisted on him -- kid like you could use some meat on his bones, then off to giggle at his joke for fifteen minutes -- then, cautiously, “Anything interesting to report?”

“No.” Yashiro catches the s that almost becomes sir. She’s exactly where she needs to be, sitting at the top of the ladder, but old habits die hard.

“You see the new graffiti in Udagawa?”

Kariya slurps at his noodles. “Yeah.” Plays with his chopsticks. “Looks like something Lord A&C would have made. We were thinkin’ he’s come back.”

You were thinking. Then I called you an idiot and told you that Arts and Crafts isn’t the only moron in this town to ever, like, do math.”

Kariya shrugs. “Dude once lectured us for two hours about eigenvectors, and you insisted on sticking around because he’d put in a good word about us to the higher-ups. You don’t ever forget the way that feels, yeah?”

“God. Thought I’d been resurrected and gone to real-kid hell.”

“So that’s our consensus.” Kariya turns back to Neku, who’d been eating quietly, thoughtfully, throughout the exchange. “A&C is back, or maybe he isn’t, and math sucks. What’s your take?”

“Dunno. Composer isn’t helping.”

(Kariya knows who the Composer is, had pieced it together in the second week of the Long Game, and he may or may not have passed that juicy bit of info on to his partner. Neither of them will tell. Neku will let them hold onto that secret, a tiny piece of privacy in the basement of the house of the soul.

Neither of them know about Angels; nobody does. Neku isn’t even supposed to know about them, but the Angels never found out about his traitorous knowledge. That’s the secret he keeps in his own personal basement -- and along with the jeans, the secret is one of the last gifts Mr. H passed on to him.)

“Hey, Phones.”

“Hm?”

“You ever read this American comic, Sandman?”

“No.”

“You should read it. You kinda remind me of the main character. -- No offense.”

“Should I take offense?”

Kariya laughs. “Y’know, I like you a lot better than the last guy. He was always going on all Composer wants this, Composer wants that, like he’d been imprinted. Never could get a straight answer out of him.”

 


 

 

Neku reads Sandman and more math appears in Udagawa on the walls and streets and lampposts: the Riemann-Zeta function, the windmill proof of Pythagoras, Fermat’s little theorem, completing the square to prove the quadratic formula. Neku spends an afternoon reading into the proof of Euler’s identity and an evening researching radian measure. Then one morning a left-side/right-side table with csc(x)+cot(x)/tan(x)+sin(x) on one side and cot(x)csc(x) on the other appears and Neku solves it with a paintbrush and silver paint; two days later it’s gone, replaced with Alhazen’s lune.

The Sandman is dour and serious and can be profoundly insensitive. He’s also meticulous and trusting and he does not go back on his word. Then something happens and he, in his own quietly contemplative way, tries to make amends with his past and present and future self. He’s presented with a choice: you accept that you’ve changed, or you don’t.

Neku decides that he’s not offended by Kariya’s comparison and that he’s glad he’s read it. (There’s a story about the dreams of a city that he particularly appreciates.) He examines Archimedes’ less than, greater than proof of the area of the circle that spills down the street up to the door of Cyco Records -- how profound that he’d been able to devise the formula and use that to calculate a numerical approximation of pi, not the other way around. The Babylonians had known about that ratio, and the Egyptians and the Greeks, and the Chinese mathematician Zu Chongzhi had drawn a shape with 24,576 equivalent sides to approximate pi to six decimal places (that is, the number 355/113), and 1200 years later Johann Lambert proved that pi is irrational: that there is no fraction of integers that can possibly express the perfect ratio between every circle’s circumference and its diameter, which is always just a little bit more than three units to one.

The set of irrational numbers is infinitely larger than the set of rational numbers, which is itself infinite -- remember Cantor. This oddly specific ratio pi (3 is the point of the 1, 4 the 1-5-9 are 2, 6-5, 3-5; 8-9, 7-9) joins hands with the rate-limiter e (2 point 7 and then 1828, 1828, 45-90-45) and all the roots of non-perfect squares in an endless set of numbers whose real names cannot be spoken, and most of its inhabitants will never be known. Imagine rational numbers as our world, and irrational numbers as every world that could ever be.

Neku finds Joshua reading Shonen Jump in the Dead God’s Pad. He works his mouth full of spit and then says, “Did you resurrect him?”

“Sorry?”

“Did you resurrect the Grim Heaper?”

Joshua furrows his brow as though he’s trying to remember, and then he looks straight at Neku. “No,” he says, with such finality that Neku is momentarily stunned into silence, “I did not resurrect the Grim Heaper.”

There’s something to Joshua’s voice when he speaks the unremitting truth, a tonal quality to it that is unlike anything Neku has ever heard -- a multitude of voices speaking in perfect harmony, perhaps, or the dreams of an entire city forced into a single mouth. Immediately upon hearing it Neku misses the sound of perfect truth and then, immediately after that, understands why the Composer takes a human form.

Joshua puts down the magazine. “Oh, Neku. You’re so charming when you get all pensive. Come here.” Neku does, reluctantly; kneels before him, and Joshua reaches out to hold him, gently, by the sides of the head, outstretched fingers around Neku’s ears like a pair of headphones.

“Certain things linger,” Joshua says. “We do not die if we are remembered.”

“Oh,” Neku says, getting it.



Gatito quietly retires the Approaching Eden set. In its place it releases a bizarre limited edition collection: a five-pin set called Approaching Infinity based on the critical points of a cosine function, starting with a pin called zero, one, and ending with two pi, one. 0-1 is by far the most powerful and rare. Collectors who do crazy things trying to get their hands on it are affectionately referred to in the communities of living and dead as stupid zeroes.

Once his legacy is set in stone the graffiti disappears. Poor Pi-Face was just fighting to not be forgotten; the Sandman’s story ends when the last page has been read; it was all just a Game, or a game, at the end of the day. Kitaniji’s old record label puts out a new remix of Twister so hot it whips the club kids into a frenzy that feels almost religious, Neku acclimatizes to the bitter taste of coffee, and thus everyone is remembered in their own way.

And what will you do, now that the story’s all wrapped up?