The Gray Avatar

Sunday morning.

Li Dong squatted on the living room sofa, his bloodshot eyes fixed on his phone screen, looking somewhat excited.

He had gnawed through University Physics until late last night, then gotten up this morning and continued grinding problems, only to run into one that left him scratching his head.

It looked like a harmless physics problem, but it was actually a mathematical demon in sheep's clothing.

[Under the influence of gravity, a particle slides from point A along a smooth curve to point B (point B is diagonally below point A and not on the same vertical line). Find the curve shape that minimizes the required time.]

The brachistochrone problem.

The physical model was simple. Anyone with even a little common sense about physics knew that a straight line was the shortest path between two points, but it definitely was not the fastest descent, because gravity had to be used for acceleration.

Li Dong's passive skill, [Orbital Calculation], let him clearly "see" the rough shape of that curve in his mind... a cycloid.

But how could he prove it?

How could he derive that analytic expression?

That was no longer something physics could solve. It required calculus of variations and solving the Euler–Lagrange Equation.

"Where's the entry point... I can't find the entry point..."

Li Dong bit his fingernail and stared at the integral sign on his scratch paper. In the end, he could only sigh helplessly.

"Forget it. Time to cheat."

He skillfully opened the "Azure Dragon Study Group."

[Currently Grinding Senior-Year Problems]: Morning, everyone. Um... my nephew ran into trouble again.

[Currently Grinding Senior-Year Problems]: He came across a brachistochrone problem. He understands the physical model, but got stuck at the final step of deriving the differential equation. I really can't spare the energy lately to derive this kind of purely mathematical stuff. Does anyone have time to give my foolish nephew some pointers?

Li Dong sent that message without so much as blushing.

For quite a while after it was sent, there was no movement in the group.

Just when Li Dong thought things were going to go awkwardly silent, an unfamiliar ID suddenly popped up.

[Johann Carl Friedrich Gauss]: The brachistochrone? The problem that little Johann Bernoulli came up with?

"Gasp—"

Gauss!

If Newton was the God of physics, then Gauss was the God of mathematics!

Ever heard of complex functions? Non-Euclidean geometry?

Never heard of any of that?

Then do you use AI? Do you know that the normal distributions all over it are also called Gaussian distributions?

Or have you heard of that classic math problem: 1 + 2 + 3... all the way to 100? That trick of pairing the first and last terms was something Gauss casually came up with in elementary school because he wanted to slack off.

Li Dong sucked in a breath. "Legendre!" He hurriedly started typing.

[Currently Grinding Senior-Year Problems]: Yes, that's the one. My nephew already handled the part where he sets up the equation using conservation of energy. He's just stuck on solving the variational problem at the end. He doesn't know how to deal with finding the extremum of the functional.

Gauss replied quickly, giving off the feeling that this was hardly worth mentioning.

[Gauss]: Heh heh, this problem does require a little calculus trick.

[Gauss]: Actually, there's no need to make it so complicated. Tell that nephew of yours that since the integrand F does not explicitly contain the variable x, he can 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?

What Gauss had said was that he did not need to explain it to Li Dong, not to his nephew.

But Li Dong did not notice that small anomaly.

Right now, his mind was filled with only one thing...

"Beltrami Identity... no explicit dependence on x..."

"That's right! The functional F(y, y') does not explicitly contain the independent variable x. According to Noether's Theorem, that means there is a conserved quantity! Using the Beltrami Identity, the originally complicated second-order variational problem is directly reduced to elementary calculus!"

Once that thin layer of paper was pierced, the equation instantly became crystal clear.

Li Dong began calculating rapidly 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 of a cycloid!

"Got it!"

Li Dong could not help letting out a low cry.

So this was Gauss? All he had to do was casually point in a direction, and all you had to do was follow the road he pointed out?

Just as Li Dong was about to thank this great master in the group for his guidance.

His phone suddenly vibrated.

A private chat window popped up.

Li Dong lowered his head to look, and his whole body froze.

[Gauss]: Interesting, little fellow. Is your side Kant's countless worlds, or Bruno's infinite universe?]

What did that mean?

Was Gauss doubting the reality of this world? Or had he discovered that he and Li Dong were not in the same spacetime?

Li Dong's mind went blank. He had no idea how to reply.

Yet during the two seconds Li Dong was stunned.

That chat window vanished into thin air!

Li Dong hurriedly looked at the "Azure Dragon Study Group."

In the group member list, Gauss's avatar had already turned gray.

"Offline status?"

WeChat did not have an offline status.

Then in this situation, Li Dong could think of only one explanation.

The group rules had been triggered!

"What the hell is going on..."

Only then did he notice some other problems.

Gauss had called Johann Bernoulli "little fellow"?

Johann Bernoulli had been dead for twenty-nine years before Gauss was even born...

And when Gauss died, how old was Beltrami? How could Gauss possibly know that this identity would be named after him?

Li Dong felt an unknown terror enveloping him.

"Knock, knock, knock..."

Just then, there was a knock at the door.

"Ah!"

Li Dong was so startled that he nearly rolled off the sofa.

He stared at the security door in horror, and an absurd thought even popped into his mind.

Could Gauss have followed the internet cable... all the way to his door?

"Who?! Who is it?!"

Even Li Dong's voice had gone shrill.

A familiar female voice came from outside.

"Junior, it's me, Lu Yao."

Hearing the words "Lu Yao," he finally breathed a sigh of relief.

Li Dong steadied himself, temporarily pushed the matter of Gauss to the back of his mind, and got up to open the door.

Lu Yao stood outside in a white cashmere coat, with a red knitted scarf making her face look especially fair.

"Junior, what were you doing in there just now? Why did your voice sound so strange?"

As Lu Yao changed shoes, she appraised Li Dong with a vaguely amused look.

Li Dong scratched his head awkwardly. "N-Nothing."

Lu Yao only felt that she had guessed something right, but she did not expose him. There was merely an extra hint of teasing in her eyes.

Li Dong felt a little creeped out under her gaze.

Why had all these girls been acting so strangely lately? Mi Xia kept turning around to look at him during class, and now this Senior Lu was doing the same. She came in without tutoring him and just kept staring at him.

Could they be after the knowledge in his head?

"Um, Sister Lu Yao, have a seat. I'll get you some water."

Li Dong hurriedly changed the subject.

"My mom isn't home 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 kind of system their supermarket uses. What era is this? They still have to do inventory manually."

"They say the system is always crashing and the data never matches up. Every weekend, everyone has to work overtime."

"The efficiency is too low, and it wears people out."

Li Dong brought over some hot water and nodded in deep agreement.

"Yeah. My mom often complains that the barcode scanner takes forever and still can't read the data. She says it'd be better to write it down by hand."

At that moment, an idea suddenly flashed through Li Dong's mind.

What if he taught himself programming and wrote them a useful inventory management system?

The more Li Dong thought about it, the more feasible it seemed.

The logic of that kind of system was not complicated. It was nothing more than database additions, deletions, modifications, and queries.

"Enough about that."

Lu Yao set down her cup of water.

"Junior, I actually didn't come to tutor you today."

Li Dong was still considering whether to use Python for the database. Hearing that, he froze.

"Huh? Then what are you here for?"

"I came to apologize."

"Apologize?" Li Dong was completely bewildered.

Women really were strange.

Lu Yao looked at his utterly confused expression and said seriously.

"Yesterday, I ran into your homeroom teacher, Teacher Yang, at a graduate school prep center."

"Teacher Yang?" Li Dong blinked.

"Mm. He teaches math there part-time, and he also happens to be my major-course advisor."

Li Dong said in surprise.

"Damn, what a coincidence. Old Yang is your teacher?"

Lu Yao nodded. "Teacher Yang told me that the Isomorphic Scaling Method used on that final differentiation problem really was something you derived yourself in class."

At that point, Lu Yao lowered her head somewhat embarrassedly.

"Last time... I misunderstood you. I thought that you..."

Only then did Li Dong suddenly understand.

"Oh, so that's what you were upset about!"

Li Dong laughed carelessly.

"Come on, what's there to apologize for? I'm a magnanimous man. I didn't take it to heart."

That response left Lu Yao a little at a loss.

"Is he joking, or does he really have no social awareness?"

Then, before Lu Yao could figure out what was going on, Li Dong spoke again.

"Sister Lu Yao, since you're not here to tutor me in high school math, could you teach me some college math?"

Lu Yao froze. "College math?"

Li Dong nodded.

"I've been teaching myself this stuff. I can understand the formulas, but the knowledge always feels scattered and disconnected."

"So I want to know, if you're a proper math major in college, what does your learning process look like?"

Lu Yao looked at the longing in Li Dong's eyes.

This time, she did not think he was aiming too high.

"Sure."

Lu Yao nodded readily and pulled her chair closer.

"Since you want to know, then Senior Sister will properly draw you this map."