Common sense cognitions that do not need to be proved are axioms, and statements that are proved true by logical limitations are theorems.

Euclid used ten axioms to derive 465 theorems, and used 465 theorems to construct the entire classical geometric world.

The geometry master that middle school students in various countries learn is actually Euclid's "Geometric Primitive" written more than 2,000 years ago.

The Geometry Originals is second only to the Bible in the world, and we are still studying geometry textbooks from more than 2,000 years ago, and may continue to do so for 2,000 years. This is the definition of "classic textbook".

Euclidean's rigorous thinking in setting axioms and then proving theorems to derive the entire classical geometric world is amazing. Even today, this kind of logical reasoning thinking and overall construction ability can be called perverted.

The only doubt is that the common-sense perception of the "Fifth Euclidean Hypothesis" is debatable.

Euclid's first 10 common sense cognitions are actually 5 axioms + 5 axioms.

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

The common-sense cognition described in Euclid's Fifth Apostulum can be expressed equivalently as follows: one point beyond a known straight line, and only one straight line can be parallel to the known straight line.

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

"I asked Sam, hey Sam, how many lines can you make parallel to a known line?"

"Sam told me that I could only make one parallel line. Sam said it was taught to him by his middle school math teacher, and he thought it was a matter of common sense, like a credit card that had to be paid on time or would be sued. ”

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

"By the way, Sam is the manager of the Tiger Lodge and the person is very nice. And Lisa is a third-year economics student, super hot, very cute, and she often goes to the Tiger Hotel to play. Speaking of which, Shen Qi drew a dot on the blackboard.

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

"That's not the point. Shen Qi looked at these otaku single dogs distressedly, and promised: "In a few months, you will be in Princeton for two years, and I will introduce you to the Tiger Hotel, where the most beautiful and exciting girls in Princeton play happily." ”

“Great-Key！”

"Great Wonder!"

The otaku chickens are frozen, and Shen Qi's image in their hearts is becoming more and more glorious, and the old man professors will not take them to the club to meet beautiful girls.

To get back to the point, Lisa, an economics major, also studies basic mathematical theories, but she doesn't think she can make more than two parallel lines beyond a straight line. Shen Qi tapped on the dot on the black and said, "What Lisa can't do, we can do for her." Euclidean's fifth axtile is common sense in most cases, but when light passes through massive planets, we have to use non-Euclidean geometry because light bends. ”

Shen Qi drew a circle with his bare hands on the blackboard, a special circle, and the circle became like this

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

"Looking at the Earth from a cosmic point of view, beyond AB, we can make an infinite number of parallel lines. ”

Shen Qi reviewed the knowledge points of Roche points and Roche straight lines for the students, and deepened the students' impression.

Professor Müller spoke very quickly in the big class, and after two big lectures, he had already touched on Riemannian geometry.

Tutorials are necessary, these second-year students are still in their academic formative years, and they need one-on-one individual tutoring, otherwise they risk going crazy.

The great Euclid left a flaw, and the universality of the Fifth Axiosum is debatable, and this is the beginning of non-Euclidean geometry.

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

Roche geometry is hyperbolic geometry, which is more from a geometric point of view. Riemannian geometry is elliptic geometry, because of the introduction of manifolds, Riemannian geometry intersects with differentiation, topology, and group theory, and it is more difficult to study.

In Roche geometry, Shen Qi deduces an infinite number of parallel lines that pass a point outside the straight line through diagramming, the concept of parallel in the traditional sense disappears, and the parallelism in curved space obeys Roche's axiom of parallelism.

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

The appearance of the triangle is no longer a traditional triangle, Shen Qi used the "devil's trumpet flower" pattern to vividly interpret the principle of the inner angle of the Roche triangle and less than 180 degrees.

Luca was a gifted student, and he drew inferences from the "Great Devil's Triangle" to illustrate that there is no such thing as a pair of incomplete, similar triangles in Roche geometry.

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

Luca is not an otaku, he has girlfriends, he has talked to several times since he came to the United States, and he is currently in an American situation, the girl is very handsome, and she is the daughter of the mayor of Anniston Township, Calhoun County, Alabama.

In Princeton, the scumbag really can't find a girlfriend.

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

Jack Ma, an Asian boy, is also an admirer of Shen Qi, who discovers that Jack Ma likes to add algebraic processing to geometric analysis.

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

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

As a gift, Shen Qi only left one homework problem, and he drew a very beautiful figure on the blackboard, such as the Eiffel Tower being reflected and displayed with the original image.

"Within three days, I hope to receive your homework, my email is written on the blackboard, you can send me the electronic version of the homework, of course, you can also submit the paper version, my office is in Room A02, 2nd floor, Luther Building, please feel free to come to me. ”

"At the beginning of the next tutorial, I will discuss this topic with you. Well, that's the end of today's tutorial, it's been the most enjoyable two hours I've had since I came to Princeton, and you're so lovely. ”

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

Twelve students fell into deep thought, staring at the beautiful and quirky patterns on the blackboard, and not a single one left the classroom for half an hour. It was already half past ten in the evening, and for these second-graders, the evening self-study had just begun.

Teaching assistants are tasked with assisting professors and guiding undergraduates through core courses.

After spending 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 scientific research.

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

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

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

The Eiffel Tower is reflected and displayed with the original image, which is a difficult problem of Riemannian geometry.

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

The other nine students made more or less misjudgments and interpretations.

Copying homework is a shameful thing at Princeton, and the consequences are so serious that it can cost a student all credibility.

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

"John, B. ”

"Al Hilal, B. ”

"Luca, A. ”

"Stephen, A. ”

"Jack Ma, A. ”

Shen Qi corrected the homework conscientiously and responsibly, and the students' tutorial homework grades should be kept on file.

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

The more Shen Qi looked at Jack Ma's homework, the happier he became, this young man can do it, he handles differential calculus in Riemannian geometry at the level of my sophomore year, I must add a + sign to him. 8)