Shen Qi stayed alone in the room, pulled up a chair, and sat down, facing the blackboard.
The task Professor Sun assigned was to supplement and improve the right diagram based on the left diagram on the blackboard.
From a mathematical logic perspective, this is not difficult to understand: derive a proof based on an assumption, or find the correct answer based on known conditions.
What is that thing in the left picture? It's a circle inscribed with a hexagon.
This is tangible geometry, namely Euclidean geometry, or at least it appears to be.
Euclidean geometry has a problem: it always aligns with people's sense of touch, but not always with their sense of vision.
Of course, this isn't a problem for over 99% of humanity. Ordinary people don't care what would happen if two parallel lines extended infinitely. I'm just taking the high-speed train home for the New Year. What does it matter to me whether the two railway tracks under the train car intersect under the definition of Non-Euclidean Geometry?
In contrast to tactile geometry is visual geometry. The former can be understood as Euclidean geometry, while the latter was called neo-geometry two hundred years ago. Lobachevsky and Riemann made the greatest contributions to neo-geometry. Today, Non-Euclidean Geometry includes Roche geometry and Riemannian Geometry.
Taking Riemannian Geometry as an example, its core idea is that any two straight lines on the same plane must intersect.
This is clearly contradictory to Euclidean geometry. According to the standard of Riemannian Geometry, any two railway tracks that extend infinitely will eventually intersect.
The classical significance of Euclidean geometry cannot be denied. In the vast universe, any civilization that has mastered basic algebra, basic Euclidean geometry, and basic laws of low-speed physics deserves to communicate, exchange resources, work together, and achieve mutual benefit. As long as those civilizations promise to abandon their research on civilian two-dimensional foil technology, we can become friends and jointly build a better home in the universe.
The perspective shifts from the vast universe back to the Milky Way galaxy - Orion Arm - Solar System - Earth - Chinese Capital - Yanjing University - a small, dark room.
The reason Shen Qi fell into deep thought was that the geometric problem on the blackboard was based on what standard: Euclidean geometry or Non-Euclidean Geometry?
Shen Qi picked up a piece of white paper from the ground and a pencil from the table. He drew a draft on the paper, copying the circular inscribed hexagon from the blackboard.
Shen Qi extends two sides AB and DE of the hexagon so that they intersect at point P.
Continue extending BC and EF until they intersect at point Q.
Extend CD and AF so that they intersect at point R.
Shen Qi connected points P, Q, and R, and muttered to himself, "Points P, Q, and R are on the same straight line... this is Pascal's Theorem?"
(Note [1] Pascal's Theorem: If a hexagon is inscribed in a circle, then the three points formed by the intersection of any two corresponding sides lie on the same straight line.)
"So this is Projective Geometry?"
Shen Qi obtained a clue, but fell into deep thought once again.
Projective Geometry is not contradictory to Euclidean geometry; rather, it is an important supplement to Euclidean geometry.
"The left figure looks like the classic graphical representation of Pascal's Theorem, so what about the right figure..." Shen Qi looked at the blackboard. The right figure showed three straight lines intersecting at point L.
Why do these three lines intersect at point L?
Is this a complete loss of the conic section, or a distorted manipulation of projection and framing?
Euclid faithfully guarded for a thousand years that parallel lines would never intersect, but why did Desargues stab him in the back, suggesting the intersection point was located at infinity?
Lobachevsky introduced Hyperbolic Geometry, to which Riemann simply smiled and said it was all bullshit. How ironic!
How many grudges and feuds are hidden behind a small blackboard?
In this mortal world, who will take over the fierce clash between old and new geometry?
Can you feel the fear of being dominated by geometry?
Click! Click!
Shen Qi tore the white paper in his hand to shreds. It was fake, all fake!
This problem has no solution; it's a fallacy!
Shen Qi scratched his head, restless. He walked to the window and drew back the curtains.
It was sunny when I left the dormitory early this morning, but now the sky is covered with dark clouds.
Projective Geometry, you bastard! You've extinguished the sun, so is today Monday?
"Who cares if it's Sunday or Monday, we don't get a break anyway. Hungry, so hungry, I get hungry as soon as I start doing problems, and even hungrier when I can't do them." Shen Qi's mouth was growling with hunger. He had just eaten breakfast an hour ago, and now he was hungry again.
"Could it be that I've activated some incredible foodie attribute?"
Shen Qi rummaged through the small room looking for food, but to his disappointment, the only edible things there were blank paper and books, all of which were spiritual nourishment.
"Never mind, I won't eat it. I'll just have to wait it out."
After a brief outburst of anger, Shen Qi forced himself to calm down. He stared at the blackboard again. Under the standard of Projective Geometry, there are many ways to complete the diagram on the right, and there is more than one correct answer.
"But Fatty Sun called me into the dark room alone. He couldn't possibly be giving me such an obvious question, could he? Even Zhou Yu'an could do this, right? He should be able to do it, right?"
For some reason, Shen Qi couldn't concentrate. His mind wandered, and he really wanted to eat a bear-shaped cookie.
I spent the whole morning lost in thought.
Around noon, the door to the small house creaked open, and Sun Erxiong, carrying a thermos in his left hand and a pot of vinegar in his right, strolled into the house humming a Northeast folk song tune.
Sun Erxiong placed the thermos and vinegar on the table and called out in a affected tone, "Young sir, your chive and shrimp dumplings are here!"
"I was quite hungry just now, but I got so hungry that I felt full again." Shen Qi shook his head and said, "Professor Sun, please have some dumplings. I'm not in a good mood because I didn't solve this problem, and I can't eat."
"Hehehe, Shen Qi, you absolutely mustn't ruin your young body like that. Food is essential for life; you'll get hungry if you skip a meal."
Sun Erxiong opened the thermos, took out dumplings, which were steaming hot and smelled delicious. "These dumplings are fresh out of the pot. You should at least eat a few. Why can't you do your homework? Why do you have to go through with this?"
"No, no, no!" Shen Qi shook his head vigorously, his gaze never leaving the small blackboard. "Professor Sun, could you at least give me a hint? Under which geometric standard should I give the answer?"
"Anything is fine." Sun Erxiong took out a few cloves of raw garlic from his pocket, cracked one open, and then ate a dumpling. "Refreshing, really refreshing. Shen Qi, you eat your dumplings first. There's no need to rush when it comes to doing problems."
"Are you a Buddhist professor?" Shen Qi picked up the chopsticks that Sun Erxiong had brought, took a dumpling, and chewed it a few times.
"Why didn't you dip your dumplings in vinegar, Shen Qi? You don't eat garlic either, what a waste of this pot of dumplings!" Sun Erxiong felt sorry for Shen Qi, who had wasted such a good dumpling.
"Hey Professor Sun, stop rambling and let's talk about something serious. You called me here to talk about math, but I've found that dumplings have become the main topic today. Hey, Professor Sun, we're getting off-topic. You need to give me a standard, otherwise I can't solve this problem."
"Shen Qi, you have many strengths, but undeniably, you also have weaknesses, at least in mathematics. You have too much and too varied mathematical knowledge and theories. If you don't have a strong ability to organize and synthesize them consistently, being too stubborn can easily lead to internal damage."
"Oh, Professor Sun, I understand these big principles. Let's get the basics sorted out first, and then delve into the deeper principles of mathematical proof, okay?"
"Hehe, let's eat dumplings, Shen Qi. That's enough." Sun Erxiong chuckled.
"Eat, I'll eat!" Shen Qi threw down his chopsticks, grabbed a chive dumpling in each hand, and started eating with his bare hands.