Free Web NovelChapter 20: Chapter 20: Gray Avatar
「Sunday morning.」
Li Dong was perched on the living room sofa, his bloodshot eyes glued to his phone screen, looking somewhat wired.
He’d been wrestling with *Big Object* until the early hours of the morning, and after waking up today, he had gone straight back to grinding through problems. That’s when he ran into one that had him completely stumped.
It was a physics problem that looked innocuous, but was actually a mathematical Demon in sheep’s clothing.
[Under the influence of gravity, a point mass slides along a smooth curve from point A to point B (where B is diagonally below A and not on the same vertical line). Find the shape of the curve that requires the shortest possible time.]
The brachistochrone problem.
The physics model for this problem was simple. Anyone with a bit of common sense about physics knew that while a straight line is the shortest distance between two points, it’s definitely not the fastest path for descent, because you have to take advantage of gravitational acceleration.
Thanks to his passive ability, [Orbital Calculation], Li Dong could clearly "see" the general shape of the curve in his mind... a cycloid.
But how could he prove it?
How could he calculate the analytic expression?
This was no longer something physics could solve. It required the calculus of variations and solving the Euler-Lagrange equation.
’The starting point... I can’t find a starting point...’
Li Dong bit his nails, staring at the integral sign on his scratch paper. Finally, he could only let out a helpless sigh.
’Forget it. Time to cheat.’
He expertly opened the "Cyan Dragon Study Group" chat.
[Senior Year Problem Grinder]: Morning, everyone. Um... my nephew’s in trouble again.
[Senior Year Problem Grinder]: He’s hit a wall with a brachistochrone problem. He understands the physics model, but he’s stuck on deriving the differential equation. I just don’t have the energy to spare for a purely mathematical derivation like this recently. Would anyone have a moment to give my foolish nephew some pointers?
Li Dong sent the message without batting an eye.
For a long while after the message was sent, the group remained silent.
Just as Li Dong thought the chat would go dead, a strange ID suddenly popped up.
[John Carl Friedrich Gauss]: The brachistochrone? Is that the problem that little fellow John Bernoulli came up with?
Li Dong drew in a sharp breath.
Gauss!
If Newton was the god of the physics world, then Gauss was the God of the math world!
Ever heard of complex analysis? Non-Euclidean geometry?
Never heard of any of it?
Well, do you use AI? Do you know that the normal distribution found all over it is also called the Gaussian distribution?
Or maybe you’ve heard of that classic math problem: 1+2+3... all the way to 100. The method of pairing the first and last numbers was something Gauss casually came up with in elementary school just to be lazy.
Li Dong gasped, then quickly started typing.
[Senior Year Problem Grinder]: Yes, that’s the one. My nephew already handled the initial step of setting up the equation using conservation of energy. He just doesn’t know how to handle the functional extremum when solving the final variational problem.
Gauss’s reply came quickly, giving off a feeling like, ’This is no big deal.’
[Gauss]: Heh, this problem does indeed require a little Calculus trick.
[Gauss]: Actually, no need to overcomplicate it. Tell your nephew that since the integrand F does not explicitly contain the variable x, he can just use the Beltrami identity (F - y’*∂F/∂y’ = C) to directly reduce the order and solve it.
[Gauss]: I don’t need to explain the rest to you, do I?
Gauss had said he didn’t need to explain it to *you*, not to *your nephew*.
But Li Dong didn’t notice this small anomaly.
His mind was now completely filled with...
’The Beltrami identity... doesn’t explicitly contain the variable x...’
’That’s right! The functional F(y, y’) doesn’t explicitly contain the independent variable x. According to Noether’s theorem, that means a conserved quantity exists! By using the Beltrami identity, the originally complex second-order variational problem is directly reduced to elementary Calculus!’
Once that final veil of confusion was pierced, the equation instantly became crystal clear.
Li Dong began to calculate furiously on his scratch paper.
Substitute into the identity... simplify... separate variables... trigonometric substitution...
x = r(θ - sinθ)
y = r(1 - cosθ)
The standard parametric equations for a cycloid!
’I got it!’
Li Dong couldn’t help but exclaim in a low voice.
’Is this Gauss? He just has to point the way, and all you have to do is follow the path he shows?’
Just as Li Dong was about to thank this great master in the group for his guidance...
His phone suddenly vibrated.
A private message window popped up.
Li Dong looked down, and then he completely froze.
[Gauss]: Interesting, little fellow. The place you’re in, is it one of Kant’s innumerable worlds, or Bruno’s infinite universe?
’What does that mean?’
’Is Gauss questioning the reality of this world? Or has he discovered that he and I aren’t in the same time and space?’
Li Dong’s mind went blank. He had no idea how to reply.
However, in the two seconds that Li Dong was stunned...
The chat window vanished into thin air!
Li Dong quickly looked at the "Cyan Dragon Study Group" chat.
In the group member list, [Gauss]’s profile picture had already turned gray.
’Offline status?’
He knew that WeChat didn’t have an offline status.
Given the situation, Li Dong could only think of one explanation.
A group rule had been triggered!
’What on earth is going on...’
It was then that he noticed some other issues.
’Gauss called John Bernoulli a "little fellow"?’
’John Bernoulli died 29 years before Gauss was even born...’
’And how old was Beltrami when Gauss died? How could he possibly know the identity was named after him?’
Li Dong felt an unknown sense of fear wash over him.
KNOCK KNOCK KNOCK...
Just then, a knock came from the door.
"Ah!"
Li Dong was so startled he nearly rolled off the sofa.
He looked at the security door in terror, an absurd thought even popping into his head.
’Don’t tell me Gauss followed my internet connection... and came to my door?’
"Who?! Who is it?!"
Li Dong’s voice was strained.
A familiar female voice came from outside the door.
"Li Dong, it’s me, Lu Yao."
Hearing the name "Lu Yao," he finally let out a sigh of relief.
Li Dong composed himself, temporarily pushing the matter with Gauss to the back of his mind, and got up to open the door.
Outside stood Lu Yao, wearing a white cashmere coat. A red knit scarf made her face seem exceptionally fair.
"Li Dong, what were you doing in there just now? Your voice sounded strange."
Lu Yao sized Li Dong up with a teasing, half-smiling gaze as she changed into slippers.
Li Dong scratched his head awkwardly. "It was... nothing."
Lu Yao felt as if she had guessed something but didn’t call him out on it, a playful look just appearing in her eyes.
Her stare was making Li Dong’s skin crawl a little.
’Why have all the girls been so weird lately? Misha keeps turning around to look at me in class, and now Senior Sister Lu comes over, but instead of tutoring, she just stares at me.’
’Could it be that they’re after the knowledge in my head?’
"Um, Lu Yao, have a seat. I’ll get you some water."
Li Dong quickly changed the subject.
"My mom’s not here today. She went to the supermarket to do inventory."
"I know, my mom went too."
Lu Yao took off her coat, hung it to the side, and sighed.
"I don’t know what’s up with their supermarket’s system. In this day and age, they still have to do inventory manually."
"They say the system is always crashing and the data doesn’t match up, so everyone has to work overtime on weekends."
"It’s so inefficient, and it really wears people out."
Li Dong brought over a cup of hot water, nodding in deep agreement.
"Yeah, my mom often complains that the barcode scanner takes forever to read the data. She says it’s faster to just write it down by hand."
As he said this, a sudden flash of inspiration struck Li Dong.
’What if I taught myself to code and wrote them a user-friendly inventory management system?’
The more Li Dong thought about it, the more feasible it seemed.
The Logic for such a system wasn’t complicated. It was basically just creating, reading, updating, and deleting from a database.
"Alright, let’s not talk about this anymore," Lu Yao said, putting down her water glass.
"Li Dong, actually, I didn’t come here to tutor you today."
Li Dong, still pondering whether he should use Python to write the database, was taken aback by her words.
"Huh? Then what for?"
"I came to apologize to you."
"Apologize?" Li Dong was completely baffled.
’Women really are strange.’
Lu Yao looked at his blank expression and said seriously,
"I ran into your homeroom teacher, Teacher Yang, at the grad school prep center yesterday."
"Teacher Yang?" Li Dong blinked.
"Mhm, he teaches a math class there part-time, and he also happens to be my advisor for my major coursework."
Li Dong said in surprise,
"Whoa, what a coincidence! Old Yang is your teacher?"
Lu Yao nodded. "Teacher Yang told me that the ’isomorphic scaling method’ used for that final, difficult derivative problem was really something you derived yourself in class."
At this, Lu Yao lowered her head, looking a little embarrassed.
"Last time... I misunderstood you. I thought..."
Li Dong finally understood.
"Oh, so that’s what you were angry about!"
Li Dong laughed magnanimously.
"Ha, what’s there to apologize for? I’m broad-minded; I didn’t take it to heart."
This response left Lu Yao at a loss.
’Is he joking, or does he really have zero social skills?’
Then, before Lu Yao could figure out what to make of the situation, Li Dong spoke again.
"Lu Yao, since you’re not here to tutor me in high school math, could you tell me about university-level math?"
Lu Yao paused. "University math?"
Li Dong nodded and said,
"I’m self-studying this stuff now. I can understand the formulas, but I always feel like the concepts are scattered and don’t form a cohesive system."
"So I was wondering, for a proper university math major, what is your learning process like?"
Lu Yao looked at Li Dong’s eager eyes.
This time, she didn’t think he was being overly ambitious.
"Okay."
Lu Yao nodded readily and pulled her chair a little closer.
"Since you want to know, then your Senior Sister will draw this map out for you properly."
