Part 4: Dodecagram

Medicine Pocket: I want to make it completely clear that I'm not asking you to solve that freakin' number problem.
Medicine Pocket: That's my job, and I'm confident I can solve it.
Misunderstandings are cleared up, and cones are bitten through.
Medicine Pocket: Can I just say this? You've got some serious tone issues you need to work on. Not everyone is as broad-minded as me, understand? Hey, are you listening to me?
The answer is no. The girl's attention is rapt entirely on the ice cream being handed to her.
Shopkeeper: Here you go, cutie pie. It's chocolate.
She holds the frosty, sweet treat in her small hand, the refreshing chill brushing past her skin.
37: This ...
Mesmer Jr.: Huh?
37: This is a perfect dodecagram! Why would it look like this?
Mesmer Jr.: You mean? Oh, looking at the shape from above. Right.
Medicine Pocket: Looks like someone's starting up again.
37: I see it! So this is some kind of mold, isn't it? What a tasteful choice!
37: I can see its shape—four equilateral triangles combined—so it can be represented by the Schläfli symbol [12/4 ...
Finding a strange satisfaction with the ice cream's shape, 37 speaks as she eagerly pulls out a notebook and pen, and starts to write.
Both notebook and pen are emblazoned with the Laplace logo. Everyone present has their own, but none have ever placed ice cream and star-shaped polygons side by side like this.
37: Let's start with 12/1—we can't skip it. Although 12/1 can be simplified to 12, so that it's actually a regular dodecagon.
37: Then there's 12/2, 12/3 ...
Five shapes appear on the paper. 37 labels each one with its Schläfli symbol and begins to evaluate them.
37: 12/3 is too clumsy, 12/5 is too extreme.
37: —And the equilateral triangle is the most stable, making the shape it forms the best of the four regular dodecagrams.
37: This is an excellent geometric shape, which means that "ice cream" must be a great dessert!
To express her appreciation of the shape, 37 draws a smiling face beneath the dodecagram labeled 12/4.
The person sitting next to her looks down at the smiling doodle and can't help but laugh.
X: Haha. A commendable choice indeed, but ...
X: Personally, I might favor this rather more charming one composed of three squares.
X pulls out his pen and draws a smiley face next to another shape.
X: I've manufactured gears in this shape before. They proved quiet and reliable in completing some of my more intriguing relays. Unassuming shapes often come with unexpected abilities.
37: I'm a bit surprised you chose 12/3. Squares are solid, simple shapes. They don't match the impression you give off.
X: So, in your impression, Miss 37, I am a restless and complex boy?
37: Well, you certainly aren't a square, but a 12/3 dodecagram is different.
37: It comes from the most basic of shapes but doesn't conform to the same rules. It's unique. I think I've picked up some clues for solving your number puzzle.
Medicine Pocket: Hey, just tell me one thing. Why do you people always aim for the low-hanging fruit? Look, 12/5 is obviously way cooler.
The childish relay continues, and a third smiley face appears, the most arrogant of them all.
Medicine Pocket: Just like an untreated thunder fig husk, prickly like a hedgehog. Its flesh isn't soft or juicy, or even edible, but it's excellent for making anti-electrical paste.
Medicine Pocket: Let me spell it out for you. Only cowards fear shapes; smart people use them.
37: You mean it's challenging, but it holds significance? That makes sense. Each interior angle of 12/5 is 30°, a special case among all acute angles.
37: Its right-angle side is always equal to half the hypotenuse. When we want a right triangle, we almost always choose between an isosceles and this.
37: What about you, Mesmer?
The question comes both inevitably and unavoidably.
Mesmer Jr.: Do I really have to choose one? Isn't this kind of juvenile? Fine, this one.
37: 12/2, a dodecagram made of two hexagons! I thought as much!
Mesmer Jr.: I just picked randomly.
37: But it feels like you. Look at these 120° interior angles. They restrain their sharpness, appearing calm and steady.
37: But hexagons are never truly stable. The collapse of any vertex can lead to catastrophic results. You have two hexagons, which helps, but they lock each other in.
37: Hmm, if either could be enlarged or reduced slightly, that would be an improvement! If they nested perfectly, we would obtain six larger, more aesthetically pleasing isosceles triangles.
Mesmer Jr.: Is this some kind of geometry-based divination? I think it's time we wrap this up.
37: Wait, we have one more shape! The regular dodecagon! We can't omit it just like that.
Medicine Pocket: I'm afraid I have to agree with you on this one, number-girl. I don't think anyone can stand having such an obvious blank space smack bang in the middle of all this.
37: But who could it be?
Mesmer Jr.: Considering where we are, should it not be Ezra?
Medicine Pocket: Well, well. Not so above our "juvenile games" as you implied.
Mesmer Jr.: Look, I just want whatever this is to end as quickly as possible.
37: I'm happy that you've come around to seeing the charm of geometry, Mesmer, but who is Ezra?
Mesmer Jr. draws another little head adjacent to 12, bestowing it with a brighter smile than her own.
Mesmer Jr.: A friend of ours. A very sincere and knowledgeable friend, and the new curator of this museum. I hope you will get a chance to meet him. I think you'd get along well.
37: Hmm. Is he a human then, not an arcanist?
Mesmer Jr.: You've heard of him?
37: No, but you said he's a dodecagon.
37: A dodecagon is different from these other shapes. It's not a star, but a regular polygon. However, a regular dodecagon is indispensable when considering twelve-vertex shapes.
37: Humans and arcanists are different. Their essence differs from ours. They are imaginary, non-existent numbers.
37: But we can still locate them using a coordinate system with i as the y-axis unit. You are a precise fraction, so if you're singing his praises, he must be exceptional. I would really like to meet him.
X: You may need a few days. He's busy preparing a special exhibit. Oh, 37!
With a gasp and a cool, unfamiliar touch, 37 is pulled back to reality from the geometric realm.
37: My dodecagram's vertex is collapsing!
X: It's melting! Quick, take a bite!
37: Nom!
Her little head collides into the ice cream.
37: This ... this is delicious!
37: With such a perfect shape, it was bound to be tasty. But it's still simply astounding!
Medicine Pocket: It's just standard ice cream churned out by a machine, though I'd say it's better than what you can get in the other refreshments areas.
Medicine Pocket: Is this your first time eating this stuff?
37: This ice cream came from that machine?
37: The doctrine tells us not to seek pleasure, nor to indulge desires, but surely bringing this machine back with me wouldn't be a total violation of the rules.
37: It would allow us to focus on studying beautiful three-dimensional shapes more joyfully. 6 would have to approve. Nom! Mmm ...
After some last-second scrambling and some well-placed bites, the melting ice cream disaster is averted.
X: So, have you found any new insights?
37: About ice cream? I've discovered that it's wonderful!
X: No, I meant your research. Have you derived any inspiration from your excursion today?
Her sky-blue eyes blink.
She looks around, then down at the ice cream in her hand, and finally slowly shakes her head.
37: I was not sure.
Mesmer Jr.: Oh, is that so?
Medicine Pocket: Come on! The setup was perfect!
Medicine Pocket: This is supposed to be the part where you realize you discovered some profound truth in the mundanity of this crappy ice cream, shout "eureka," and then rush back to the lab to work on your breakthrough.
Mesmer Jr.: Medicine Pocket, you don't really believe that, do you? No one at Laplace would laugh at that joke.
37: The exhibits in this museum are all very interesting, and this ice cream is delicious. I've found many good numbers today.
37: But ...
The girl looks down at her hand. Where her perfect dodecagram had been, but now nothing remains.
Then she looks up.
Upon the real-time star map on the museum ceiling, stars twinkle beautifully. A rare chance to appreciate the grandeur of the cosmos with the naked eye.
The paths of the planets are clearly marked, and the vast Milky Way is unobstructed by light or atmospheric pollution. Every pixel is perfectly placed.
37: What I do at the computing center seems to be the same. I came to get more numbers, talked with people from different fields, and it brought me these in return.
37: I think I've grown somewhat. I've seen things, learned more, and gained more numbers than when I was on the island.
37: However ...
37: When I look down, I still can't see where the next square is.
Mesmer Jr.: Oh, is that so?
This is not the expected result of her treatment.
Mesmer Jr. suddenly recalls the conjecture about planetary distances and Platonic solids that the girl once marveled at in the lab.
Johannes Kepler, hailed the "legislator of worlds," discovered the three laws of planetary motion. In his book, Mysterium Cosmographicum, he proposed a hypothesis.
He claimed that the distances of the six planets that had been known at the time—Mercury, Venus, Earth, Mars, Jupiter, and Saturn—from the sun could form five perfect Platonic solids.
The arrangement fit so perfectly. Deep down, even the atheists of the time hoped that God had arranged the universe according to such a delicate blueprint.
But as we know now, his hypothesis was incorrect.
Mesmer Jr.: ...
Mesmer does not fully understand 37's goals or her methods. But she astutely observes that 37's research and all her efforts are predicated on a shared hypothesis.
A hypothesis as beautiful as Kepler's conjecture, but equally untrue.
Mesmer Jr.: Have you considered ...
That your hypothesis might be wrong?
37: What?
Mesmer Jr.: Nothing.
37: Perhaps I was a little hasty.
37's gaze shifts away from the invisible square on the floor.
37: John Napier took twenty years to devise logarithms, and Fermat's Last Theorem took 358 years to be fully proven. It's likely that my answer won't be found overnight, either.
37: I know that I need to hurry. I can't afford to waste time. But ...
37: The doctrine instructs us never to lose composure or reason, to adhere to the laws of all things and ascend the divine steps, so we can reach the highest height, and not descend into the abyss of chaos.
37: I shouldn't forget my limitations.
The other three exchange glances.
Medicine Pocket: Congratulations! It seems our mathematician has finally discovered a basic truth.
Medicine Pocket: Great achievements don't come overnight. The more challenging your research, the more time and effort it will take. These things are proportional.
X: Your research is fascinating, and I look forward to the day it is completed, but don't overexert yourself. That's just a little bit of advice from your senior at the computing center.
X: Many talented researchers have focused excessively on their work, forgetting they too are mortals. We don't want you to follow in their footsteps, 37.
37: Hmm? I don't know if I understand what you mean. I'm feeling just fine.
Mesmer Jr.: I thought as much.
X: Wow, this is worse than I anticipated. Look ...
X chooses his words carefully, but Medicine Pocket just rolls their eyes and chimes in.
Medicine Pocket: You look like a deflated bloodmoon jellyfish, oozing a vibe like you've been stuck in a plastic bottle without a water change for days. You've got tentacles floundering all over the place!
Medicine Pocket: And I've been meaning to ask: What's with the messy hair? Even the cactus cat in my lab grooms itself better than you.
37 looks around in confusion, attempting to locate her fictitious tentacles floundering in the air.
37: Oh, I see. It's a metaphor, isn't it?
Medicine Pocket rolls their eyes again in response.
Mesmer Jr.: Whatever else you do, try not to stay cooped up in that lab for too long. People need a little entertainment to keep themselves sane, and that's advice from a therapist.
Mesmer Jr.: I already have enough work on my hands. I don't want to see you lining up at the door of the artificial somnambulism therapy room.
37 blinks, contemplating for a moment, before fixing her bright eyes on the group.
37: Are you all really so concerned about me?
Someone takes issue with her choice of words.
Medicine Pocket: Yeah, right! Like how nuts are we for caring if you work yourself into madness?! You're already beyond delusional.
37: You're all so kind. Thank you.
37: I don't feel like I'm pushing myself, but spending time with all of you makes me happy. It reminds me of the past.
37: Whenever my friend Sophia would come to see me, if I was working on a problem, she'd linger at the door for the longest time before coming in.
37: She'd always ask, "Am I disturbing you, 37?"
37: To tell the truth, her visits did sometimes slow down my progress.
37 shakes her head.
37: But research can always be continued later. The truth is, I was happy she came to see me.
X: It sounds like you have a good friend there.
37: Yes. She ... She's not here now ... but I hope I can introduce her to everyone someday.
X: I'd be honored to share this secret base of ours with you and your friend.
Their time together seems to be nearing its natural close.
X: Speaking of which, remember the special exhibit I mentioned?
X: It opens in three days. Our friend Ezra—our dodecagon—exerted considerable effort to facilitate it. Medicine Pocket and I assisted him a little too.
X: If you have time, you'd be very welcome to come and join us as we view its offerings. We can peruse the exhibit together. Of course, if our dear therapist would like, she can join us, too.
37: Thank you for the invitation. I'd like to come. Then I can have another ice cream.
Medicine Pocket: You really like that junk, huh?
37: I just appreciate its geometry. It's nothing related to its taste. I think it helps my mind work more efficiently.
Medicine Pocket: That's the lamest excuse I've ever heard! So if I gave you a scoop of anti-electric paste that looked like ice cream, would you eat that too?
Mesmer Jr.: Give it a rest? Don't you find this line of questioning to be a little childish?
Medicine Pocket: Why are you always taking sides? And worse, never mine!
Their farewell chatter seems to go on without end. The young researchers debate over the essence of things, and the functions of the taste and digestive systems.
Until, at last, the fifth party present brings an end to their interminable digressions.
Shopkeeper: Uh ... sorry, everyone, excuse me.
Medicine Pocket: Huh?
Medicine Pocket seems to have genuinely forgotten the ice cream shopkeeper was there.
Shopkeeper: It's just, I heard you say "next time," and, I'm glad you enjoyed yourselves, but ... I'm afraid there won't be any more ice cream next time.
Shopkeeper: My shop, well, it's closing down.