kscans

Chapter 61 - Chapter 2: Part 2, The Power of Numbers

"That's interesting, but first, I want to know the relationship between size and cost using the simplest processing method."

I said this while glancing at Mia. She had been scowling just a moment ago, but now she was listening to the explanation with rapt attention.

"It's not that simple. As the size increases, the rate of increase for the required magic power also goes up. Does that make sense?"

"I see. What are the specific numbers?"

"What would you do with that information anyway? It obviously changes depending on the alchemist's aptitude, physical condition, and other factors. Under normal circumstances..."

Noel wrote down five values. Apparently, they represented the magic power required for 1 unit, 2 units, 5 units, 10 units, and 100 units. For reference, creating a single sword required around 100 units.

"Please."

I handed her the paper. Mia's eyes lit up.

"When the volume increases by a factor of n, the required magic power per unit of volume increases by n to the power of 1.2."

"I see. That's a bit steep."

"Wh-what are you talking about?"

It seemed she couldn't understand my conversation with Mia.

"I'm just guessing, but you meticulously shifted the values you wrote down by plus or minus five percent from the actual required amount. You should be careful, a disguise with such a clear pattern is the same as having no disguise at all."

"For Mia, anyway. By the way, what about you?"

"…………"

Noel wrote down five new numbers. Her handwriting was a little shakier this time.

"N to the power of 1.1. It seems you are indeed quite skilled."

"That's amazing."

"Y-you don't understand any of this, do you? Your bluffs won't work on me."

"You can process 32 units of magic metal with two-thirds of the magic power a normal alchemist would need. For 64 units, you'd need a little under sixty percent, and for 128 units, you could do it with just over half."

"N-no way. How do you know that?"

Apparently, I was correct. Fulsy's sales pitch hadn't been an exaggeration.

"I just calculated it from the numbers you wrote down."

"...Is this some kind of merchant technique? Hmph. Well, it seems you understand how amazing I am."

Having recovered from her surprise, Noel puffed out her chest proudly. Uh oh, Mia's gaze sharpened. Well, whatever. I'll let her have this moment so things go smoothly.

"I understand the size issue. Next is shape. Specifically, how does alchemy process magic metal?"

I looked around the room. I couldn't see anything that looked like alchemy equipment.

"...To give you a practical demonstration, I'd have to use a magic crystal, and with my current authority, I can't afford to waste it. I can't exactly say I used it to show a merchant."

"I have some that I received from the director. From what you just said, a piece less than one unit shouldn't require that much magic power, right?"

I took out a fragment of a magic crystal.

"A magician entrusted this to a commoner..."

Noel brought out a trapezoidal stand with a hemispherical dish on top. She carefully placed the magic crystal I gave her into the dish. Then, in the center of the stand, on a spot with a circular pattern, she placed a piece of magic metal about a quarter of the size of the previous one.

"Watch this. This is alchemy."

When she placed both hands on the stand, the magic crystal glowed red. Light traced along the patterns drawn on the stand. Finally, the magic metal emitted a golden radiance. The next moment, the square piece of metal had become a perfect sphere.

"That's incredible. So you can create any shape you imagine?"

The mystery was greater than I had imagined, and I was genuinely surprised. This is like having a CAD system and a precision processing machine all in one.

"So you understand. By the way, you can't just make any shape you want. There's a very strict theory behind it."

Noel began to explain the principles of shaping things with alchemy. The "strict theory" was surprisingly easy to understand.

"In other words, you can only make shapes that can be drawn with a compass and a ruler."

"…………What are you? Are you really a commoner? Don't tell me information from the Magic Division has been leaked."

Noel shook her head at Mia's words. So Mia was right again. Of course, in practice, she was creating a three-dimensional object. It was geometric construction in space, not on paper.

"I'm guessing, but the efficiency of the magic probably worsens with complexity, right? For example, even if you draw lines of the same length, a straight line and a circle would require different amounts of magic power."

"Th-that's right. Unlike a straight line, which can be defined once its length is set, a circle requires an outrageous amount of magic power."

Definition, huh? Then there's hope. Mia and I nodded at each other.

"So how does it work, specifically?"

"………………There's no way you'll understand this, but I'll tell you. You multiply the circle's diameter by a special number."

"By the way, what is that number?"

"As if I'd tell you. It's a secret of alchemy."

"You could at least tell me how many digits it has. I mean, worst case, you could just put ink on a wheel and roll it on a piece of paper..."

"Three digits!"

Noel said, as if in desperation. Wow, it's that primitive? Still, they must have worked incredibly hard to get three digits using that method. I've won this round. There's no guarantee it's the same in this world as it was in my old one, but it should probably be fine. I can't imagine that creatures from Earth, let alone matter itself, could exist in a world where things like the natural logarithm are different.

Besides, if my theory is correct, I can test it right away.

"3.14, huh..."

"…………where from?"

"Well, that number is just the ratio of a circle's circumference to its diameter, isn't it?"

It's what's known as pi, one of the most, if not the most, important numbers in mathematics, and maybe even the world.

"If the precision of π (pi) improves, does the required magic power decrease? Logically, if the circumference is determined, the amount of information in that line is the same whether it's a curve or a straight line."

Of course, pi never becomes a "normal" number in the usual sense. It's a transcendental number, after all. It takes an infinite number of digits to write it down.

In any case, if magic is a form of super-efficient information processing like I imagine, it should have a great affinity with mathematics.

"Alright, what if π (pi) was 3.142?"

"Don't use such strange words... 'Circlet' is 3.142, you mean. Hmph, fine. I'll give it a try."

Noel turned the sphere she had just made back into a cube. I see, it reverted in an instant. That must be how different straight lines and curves are. Then, she performed the same process again.

"No, this can't be..."

"By how much was it reduced?"

"...About eighty-five percent."

Noel said with a look of someone who had just witnessed the impossible. With a sphere, the volume involves pi cubed. It's not surprising that a small difference creates a huge one.

"Okay, next is 3.1416."

Noel twitched at Mia's words. Still, she repeated the process.

"A-a little over seventy percent. What is this? What am I seeing right now?"

"3.141592."

"It's been halved..."

"Alright, then,"

"Wait. Any more than this, and I won't be able to make the adjustments myself. But tell me. Tell me, how did you do it?"

"You can calculate the circumference ratio like this."

Mia drew a circle on a piece of paper with a regular polygon inscribed within it. As the number of corners on the polygon increases, the total length of its sides gets closer to the circumference of the circle. When I taught Mia this principle, she calculated it all the way up to a 128-sided polygon. The truth is, though, that's not even enough to achieve six decimal places for pi.

The only reason I know pi to six decimal places is because a former teacher of mine, fed up with a simplified education system, made his seminar students memorize it to ten digits. I've forgotten the last few, as expected.

Incidentally, even if you increase the number of sides on the polygon to infinity, it never becomes exactly the same as the circle. But I am a little curious what would happen if you defined it accurately using the concept of an infinite series. Well, the only one I know is that one that goes 1 minus 1/3 plus 1/5 minus 1/7... just alternating plus and minus with odd numbers. I think it takes tens of billions of calculations to converge on a decent value for pi.

"How do you know something like this?"

Noel looked ready to grab Mia. There's no point in me acting all high and mighty when I'm merely standing on the shoulders of giants.

This must be the drawback of having one person handle both design and manufacturing. They inevitably get bogged down by the practical work. The fact that alchemists are such a valuable practical resource probably keeps them from pursuing abstract mathematical theories.

It's like Archimedes trying to craft the king's crown himself to run his experiment.

By the way, there are probably no actual giants in this world. I already checked to see if Neanderthals existed here, perhaps being called orcs or something. There were no orcs, even though female knights exist.

"So, what happens if you make my order with this new standard?"

"...I can make one. But didn't you say you wanted a lot of these? How many?"

"Ultimately, several thousand or more."

"Hah, so your true colors are showing. We wouldn't have enough magic crystals even if we gathered every last one in the kingdom."

Noel was back to her energetic self. I scratched my head.

"Ah, I forgot to explain. What I actually need you to make isn't the spheres themselves, but a mold for the spheres. Six of those, and also these other two shapes, all within a piece of magic metal about the size of a brick..."

I spread out the blueprints I had drawn. They depicted a total of three different types of molds. I actually needed one more, but a blacksmith should be able to handle that one. And the material for the final product isn't magic metal.

"These all use circles. This would normally be impossible."

"So, what's the verdict?"

"With this new circlet... or whatever... I can manage. But..."

"What are you missing?"

"I'll need a fairly large medium-sized crystal. At our facility, even the director has to submit a formal request to use one of those."

"Got it. I'll figure something out."

I'm pretty sure there was one among the spoils from the Third Knight Order, now the Monster Knight Order. Apparently, dragons have multiple magic crystals, not just in their foreheads but also their hearts and elsewhere. I'll have to consult with that prince. Well, it's important for them too.

"Alright, first, I need you to make a one-third scale prototype of this."

"I can do it with the magic crystal I have on hand, but..."

"I'll talk to Fulsy about getting you a replacement. So what do you say, will you take the job?"

"...Fine, but..."

Noel looked at Mia.

"Mia, could you help her out a little?"

"If you're asking, senpai, I suppose I have no choice."

…………

"No, that's not right. If you draw an auxiliary line like this, you'll get the answer."

"I-I see..."

"That angle and this angle are the same, because it's proven they are similar."

"True, but that's no good. It increases the required magic power. You can't let the lines cross unnecessarily."

"Then use this to reduce the number of calculations."

"That might work, but then this part here..."

Mia and Noel were deep in discussion over the blueprint I gave them. I was left with nothing to do.

"Sorry to interrupt, but I'd like to know how long the prototype will take."

I ventured to ask the two, who were in the middle of a heated debate.

"Wait, we're at a really good part."

"Hold on, what you're doing right now has nothing to do with my request, does it? That's just some random geometric drawing."

"...I-I got carried away."

"You too, Mia... Anyway, at the very least, just the sphere part is fine. That's the most important for checking the precision, and the prototype casing can be made of wood."

I was exasperated. They were at each other's throats just a little while ago. But I suppose this was to be expected. It seems Mia's geometric sense and an alchemist, this world's equivalent of a CAD designer, are a good match.

"...You're right. Including the time for the magic to dissipate and the shape to set... I'd say three days."

"Infinity has become finite," Mia said with a mischievous smile. After all, her initial response had been "absolutely impossible."

"…………w-what is it? You're certainly amazing, I'll admit, but still..."

"Well, that's true. We can't do the practical side. So, on that note."

"What?"

"When I show this to the other members, I'd like you to be there."

"Other members... you mean merchants, right?"

"Uh, well, that's most of them, yes. But the director will be there too. He'd probably be a pain if I didn't invite him."

"I'll go."

She answered instantly.

"After that, you'll have to talk with blacksmiths and such when we go to actually produce it, but..."

"H-hmph. Compared to being taught mathematics by a commoner, nothing fazes me anymore. S-so... um, from now on..."

Noel looked at Mia.

"If you can finish not just the sphere but the other two parts in three days, I wouldn't mind lending a hand."

"Hey, I'll run out of magic power myself, you know. ...Fine, I'll do my best."

"Hey, Mia. You don't have to push her that hard. The people I'm showing the prototype to are who they are, and I don't even know when the meeting for the trade fair will be."

I said to Mia. But she was already locked in a spirited back-and-forth with Noel.

Well, all's well that ends well, I guess. But Noel, she's my precious secretary. Don't you forget it.

Still, magic really is a form of super-efficient information manipulation. I might be able to think about the mechanism of the Oracle's crystal in the same way.