Common sense cognition that does not require proof is an axiom, and statements that are proofed by logic are theorems.

Euclid used ten axioms to derive 465 theorems and constructed the entire classical geometric world with 465 theorems.

The geometric masterpiece learned by middle school students from various countries is actually the "Geometric Origin" obtained by Eucli more than two thousand years ago.

The number of publications and distributions of "Geometry Origin" worldwide is second only to the Bible. We are still studying geometric textbooks from more than two thousand years ago, and perhaps we will continue to study for two thousand years. This is the definition of "classic textbooks".

Euclid first sets the axiom and then proves the theorem, thus deducing the entire classical geometric world is amazing. Even today, this logical reasoning thinking and overall construction ability are perverted.

The only doubtful point is that the common sense cognition of "Euclid's Fifth Public" is worth discussing.

The 10 common sense cognitions that Euclid set first are actually 5 axioms + 5 public assumptions.

Modern mathematics combines the assumptions and axioms, and the cognition of common sense is collectively called axioms.

The common sense cognition described in "The Fifth Assumption of Euclid" can be expressed equivalently as: one point outside the known straight line can be made and can only be parallel to the known straight line.

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

"I asked Sam, hey Sam, how many straight lines can be made one point outside the known straight line parallel to the known straight line?"

"Sam told me that he could only make a parallel line. Sam said that this was taught by his middle school math teacher, and he thought it was a common sense problem, just like a credit card must be paid on time, otherwise he would be sued."

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

"By the way, Sam is the manager of Tiger Hotel and is very good. Lisa is a third-year economics student. She is super hot and cute. She often goes to Tiger Hotel to play." Speaking of this, Shen Qi drew a dot on the blackboard.

"Wow, Qi, have you dated Lisa?" Half of the twelve math boys were otaku and had no girlfriend, and they felt envious.

"This is not the point." Shen Qi looked at these otakus single dogs with distress and promised: "In a few months, you have been in Princeton for two years. I will introduce you to the Tiger Hotel. Princeton's most beautiful and exciting girls are all there to have fun."

“Great-key!”

"Great wonder!"

The otakus were frozen, and Shen Qi's image in their hearts became more and more magnificent. The old professors would not take them to the club to meet the beautiful girls.

"Back to the point, the focus is that Lisa, a major in economics, also learns basic mathematical theories, but she does not think that if she goes outside the straight line, she can make more than two parallel lines." Shen Qi knocked on the small dot on the black and said: "We can do what Lisa cannot do for her. The fifth post of Euclid is common sense in most cases, but when light passes through a massive planet, we must use non-Euclidean geometry because the light will bend."

Shen Qi drew a circle with his bare hands on the blackboard, a special circle, and this is how it is.

Two points a and b in the circle are connected by a curve, which represents a straight line in non-European geometry.

"Looking at the earth from the perspective of the universe, we can make countless parallel lines a little beyond the ab."

Shen Qi gave students a review of the knowledge points of Roche Points and Roche Lines to deepen their impressions.

Professor Mueller gave a quick lecture in the big class, and after two big classes, he talked about Riemann geometry.

Tutoring classes are necessary. These second-year students are still in their academic growth stage and need one-on-one separate tutoring, otherwise they may go crazy.

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

Non-European geometry is divided into two main schools, Robachevsky geometry and Riemann geometry.

Roche geometry is hyperbolic geometry, and it starts more from a geometric perspective. Riemann geometry is elliptical geometry. Because of the introduction of manifolds, Riemann geometry intersects differential, topology, and group theory, and it is more difficult to study.

In Luo's geometry, Shen Qi interpreted infinitely many parallel lines that were a little outside the straight line through drawing. The traditional concept of parallelism disappeared, and parallelism in the curved space complied with Luo's parallel axiom.

In this special space, the sum of the inner corners of the triangle is no longer 180 degrees, but is less than 180 degrees.

The appearance of the triangle is no longer a traditional triangle that is upright. Shen Qi uses the "Devil's Horse Flower" pattern to vividly interpret the principle of the inner corners of the Roche triangle and the principle of less than 180 degrees.

Luca is a very talented student. He learned from others and made a set of "big devil triangles" to interpret the existence of any pair of incomplete similar triangles in Roche's geometry.

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

Luca is not an otaku. He has a girlfriend. He has talked about several times after coming to the United States. He is currently a partner in the United States. The girl is quite handsome and is the daughter of the mayor of Aniston Town, Calhoun County, Alabama.

In Princeton, the poor student really couldn't find a girlfriend.

Only when the school is done can boys have the energy and time to socialize, participate in sports and show their muscles, thereby attracting and meeting beautiful girls.

Jack Ma, an Asian boy, is also someone Shen Qi admires. Shen Qi found that Jack Ma likes to add algebraic processing to geometric analysis.

Jack-Ma explained the geodesic linear trait problem in the normal curvature space through a beautiful two-dimensional manifold, which was praised by Shen Qi.

The academic atmosphere in the guidance class was quite enthusiastic. Shen Qi won the respect and favor of all students with 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, like reflecting the Eiffel Tower and displaying it with the original image.

"Within three days, I hope to receive your homework. My email address was written on the blackboard. You can send me an electronic version of the homework, and of course you can submit the paper version. My office is in Room A02, 2nd floor, Luther Building. Welcome to come to me at any time."

"I will discuss this topic with you at the beginning of the next guide course. Okay, today's guide course ends here. This is the most enjoyable two hours I have spent since I came to Princeton. You are very cute."

Shen Qi smiled and waved his hand and left the classroom.

The twelve students fell into deep thought. They stared at the beautiful and weird patterns on the blackboard. No one left the classroom within half an hour. It was already ten:30 in the evening. For these second-grade students, 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 cute big boys from all over the world, Shen Qi devoted himself to his main business, and his main line was of course the scientific research line.

The proof work of "Müller-Shen Theorem" is implemented as planned.

Shen Qi completed a small detail process in three days, proving that each weak* closed convex set is a near-force set, so that ‖x*-yn*‖→dist(x*,a*)(n→∞) is not lost in generality.

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

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

Only three students successfully proved using the Gauss-Bone theorem that the Eiffel Tower and its reflection were curved spaces.

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

Copying homework in Princeton is a shameful thing, and the consequences are so serious that a student will lose all his reputation.

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

“John, b.”

“Arshilar, b.”

“Luca, a.”

“Stephen, a.”

"Jack Horse, a."

Shen Qi corrects homework seriously and responsibly, and the students' homework results of the guided courses must be archived.

"Jack Ma, I'll take a look again, um, a+."

The happier Shen Qi looked at Jack Ma’s homework. This young man can do it. His differential treatment in Riemann geometry is as good as that of my sophomore year. I have to add a + sign to him.
Chapter completed!