I Just Want to Be a Quiet Top Student

Common-sense cognitions that do not require proof are axioms, and statements that are true when proven to be true are theorems.

Euclid derived 465 theorems with ten axioms, and constructed the entire world of classical geometry with 465 theorems.

The geometry master model learned by middle school students in various countries is actually the "Elements of Geometry" written by Euclid more than 2,000 years ago.

The number of publications of "Elements of Geometry" worldwide is second only to the "Bible". We are still studying geometry textbooks more than 2,000 years ago, and may continue to study for another 2,000 years. This is the definition of "classic textbook".

Euclid first set the axioms and then proved the theorems, thus deriving the rigorous thinking of the entire classical geometric world is amazing. Even today, this kind of logical reasoning thinking and overall construction ability can be called abnormal.

The only doubt is that the commonsense cognition of "Euclidean fifth postulate" is debatable.

The 10 commonsense cognitions first set by Euclid are actually 5 axioms + 5 postulates.

Modern mathematics combines postulates and axioms, and commonsense cognitions are collectively called axioms.

The common-sense cognition described in the "Fifth Postulate of Euclid" can be equivalently expressed as: Passing through a point outside the known straight line, one and only one straight line can be drawn parallel to the known straight line.

"The fifth postulate is not difficult to understand, and it has become the common sense of most people in the society." Shen Qi casually drew a straight line with a length of 1.8 meters on the blackboard. It was very straight, as if compared with a ruler.

"I asked Sam, Hey Sam, how many straight lines can be drawn parallel to the given line through a point outside the given line?"

"Sam told me that only one parallel line can be drawn. Sam said this was taught by his middle school math teacher. He thinks it is a common sense problem, just like a credit card must be paid on time, otherwise it will be sued."

"I asked Lisa and she gave me the same answer."

"By the way, Sam is the manager of the Tiger Hotel. He is very nice. And Lisa is a third-year economics student. She is super hot and cute. She often goes to the Tiger Hotel." Having said that, Shen Qi wrote on the blackboard A dot is drawn on it.

"Whoa, Qi, you're already dating Lisa?" Half of the twelve math department boys are nerds and don't have girlfriends, and they feel envious.

"That's not the point." Shen Qi looked at these nerd singles with distress, and promised: "In a few months, after you have entered Princeton for two years, I will introduce you to the Tiger Hotel. The most beautiful and hottest girls in Princeton Have fun there."

"Great-Key!"

"Great wonder!"

The homeboys were freezing, and the image of Shen Qi in their hearts became more and more glorious. Old men and professors would not take them to the club to meet beautiful girls.

"Let's get down to business, the point is that Lisa, who majored in economics, also studies basic mathematical theory, but she doesn't think that if she goes a little outside the straight line, she can draw more than two parallel lines." Shen Qi tapped the dot on the blackboard and said to: "What Lisa cannot do, we can do for her. Euclid's fifth postulate is common sense in most cases, but when light passes through a massive star, we must use non-Euclidean geometry because the light will bend."

Shen Qi drew a circle on the blackboard freehand, a special circle, and it turned out like this ○

The two points A and B inside the circle are connected by a curve, which represents a straight line in non-Euclidean geometry.

"Looking at the earth from the perspective of the universe, we can draw countless parallel lines beyond AB."

Shen Qi reviewed the knowledge points of Roche points and Roche lines for the students to deepen the students' impression.

Professor Muller spoke very quickly in the big class. After two big classes, he had already talked about Riemannian geometry.

Tutorial classes are necessary. These second-year students are still in the stage of academic growth. They need one-on-one individual tutoring, otherwise they may go crazy.

The great Euclid left a loophole, the universality of the fifth postulate is questionable, and this is the beginning of non-Euclidean geometry.

Non-Euclidean geometry falls into two main schools, Lobachevsky geometry and Riemannian geometry.

Roche geometry is hyperbolic geometry, more from the geometric point of view. Riemannian geometry is elliptic geometry. Because of the introduction of manifolds, Riemannian geometry intersects with differential, topology, and group theory, making it more difficult to study.

In Rochester's geometry, Shen Qi deduces infinitely many parallel lines passing through a point outside the straight line through drawing. The concept of parallel in the traditional sense disappears, and the parallel in curved space obeys Rochester's parallel axiom.

In this special space, the sum of the interior angles of a triangle is no longer 180 degrees, but less than 180 degrees.

The shape of the triangle is no longer the upright traditional triangle. Shen Qi used the pattern of "devil's morning glory" to vividly interpret the principle of the interior angle of Roche's triangle and its smaller than 180 degrees.

Luca is a very talented student. He made a set of "big devil triangles" by analogy, explaining that there is no pair of incongruent similar triangles in Roche's geometry.

Italians seem to have innate inspiration and intuition for drawing, painting, and architecture. Shen Qi appreciates Luca, an Italian student. Luca is talented, has a distinctive image, and maintains the usual aggressiveness of Italian men.

Luca is not an otaku. He has a girlfriend. He has talked with him several times since he came to the United States. He is currently in an American relationship. The girl is quite handsome. She is the daughter of the mayor of Anniston, Calhoun County, Alabama.

In Princeton, scumbags really can't find girlfriends.

Only under the premise of finishing their studies, boys have energy and time to socialize, participate in sports to show their muscles, so as to attract and meet beautiful girls.

Jack Ma, an Asian boy, was also admired by Shen Qi. Shen Qi found that Jack Ma liked to add algebraic processing to geometric analysis.

Jack Ma explained the geodesic shape problem in constant curvature space through a set of beautiful two-dimensional manifolds, which was praised by Shen Qi.

The academic atmosphere in the tutoring class was relatively warm. Shen Qi won the respect and favor of all the students with his profound mathematical knowledge and easy-to-understand explanation methods.

As a meeting ceremony, Shen Qi left only one homework question. He drew a very beautiful figure on the blackboard, as if the Eiffel Tower was processed as a reflection and displayed together with the original image.

"Within three days, I hope to receive your homework. My e-mail address is written on the blackboard. You can send me an electronic version of your homework, of course you can also submit a hard copy. My office is on the second floor of the Luther Building Room A02, please come to me anytime.”

"At the beginning of the next tutorial, I will discuss this topic with you. Well, this is the end of today's tutorial. This is the most enjoyable two hours I have spent in Princeton. You are very lovely. "

Shen Qi smiled and waved, and left the classroom.

Twelve students fell into deep thought, staring at the beautiful and weird patterns on the blackboard, none of them left the classroom within half an hour. It was already 10:30 in the evening. For these second-year students, the evening self-study had just begun.

The task of teaching assistants is to assist professors and guide undergraduates to complete core courses.

After spending a pleasant two hours with a group of lovely big boys from various countries, Shen Qi devoted himself to his main business, and his main line was of course scientific research.

The proof of the "Muller-Shen Theorem" was carried out as planned.

Shen Qi completed a small detail processing in three days, proving that each weakly*closed convex set is a close set, so that ‖x*-yn*‖→dist(x*, A*)(n→∞) is not lost general.

Today is the deadline for Shen Qi to collect homework, and all twelve homeworks have been handed in.

After the Eiffel Tower is reflected and displayed together with the original image, this Riemannian geometry problem is somewhat difficult.

Only three students have successfully proved, using the Gauss-Bonnet theorem, that the Eiffel Tower and its reflection are curved spaces.

The other nine students made more or less wrong judgments and interpretations.

Copying homework is a shameful thing in Princeton, and the consequences are so serious that it can completely destroy a student's credibility.

Shen Qi believed that his boys would not copy homework, so he corrected twelve homework one by one.

"John, B."

"Al Hilal, B."

"Luca, A."

"Stephen, A."

"Jack-Ma, A."

Shen Qi corrects homework seriously and responsibly, and the students' homework grades for tutorial courses must be kept on file.

"Jack Ma, I'll take another look, um, A+."

The more Shen Qi looked at Jack Ma's homework, the happier he was. This guy is fine. His differential processing in Riemannian geometry is at the level of my sophomore year. I must add a + sign to him.