You're All So Cute
Common sense knowledge that requires no proof is called an axiom, and statements that are true after being proven by logical constraints are called theorems.
Euclid derived 465 theorems from ten axioms, and constructed the entire world of classical geometry with these 465 theorems.
The geometry model studied by secondary school students in various countries is actually Euclid's "Elements," written more than two thousand years ago.
Elements has been published and distributed in the world second only to the Bible. We are still learning geometry from a textbook from more than two thousand years ago, and perhaps we will continue to learn it for another two thousand years. This is the definition of a "classic textbook".
Euclid's rigorous thinking in first setting up axioms and then proving theorems, thereby deriving the entire world of classical geometry, is astonishing. Even today, this logical reasoning and overall construction ability is nothing short of extraordinary.
The only point of contention is that the common-sense understanding of "Euclid Fifth Postulate" is questionable.
Euclid's original ten common-sense principles were actually five axioms plus five postulates.
Modern mathematics combines postulates and axioms, collectively referring to common-sense knowledge as axioms.
The common-sense knowledge described by Euclid's Fifth Postulate can be equivalently expressed as: Through a point outside a given line, one and only one line can be drawn parallel to the given line.
"The Fifth Postulate is not difficult to understand and has become common knowledge for most people in society." Shen Qi casually drew a straight line 1.8 meters long on the blackboard, it was very straight, as if it had been measured with a ruler.
I asked Sam, "Hey Sam, how many lines can be drawn through a point outside a given line that are parallel to the given line?"
"Sam told me that only one parallel line can be drawn. Sam said that's what his high school math teacher taught him, and he considers it common sense, just like credit card bills must be paid on time or you'll be sued."
I asked Lisa, and she gave me the same answer.
"By the way, Sam is the manager of Tiger Inn, and he's a really nice guy. Lisa is a third-year economics student, super hot and cute, and she goes to Tiger Inn a lot." Shen Qi drew a dot on the blackboard as he said this.
"Wow, Qi, you're already dating Lisa?" Half of the twelve Mathematics Department majors were nerdy and didn't have girlfriends, and they were envious.
"That's not the point." Shen Qi looked at these single guys with pity and promised, "In a few months, after you've been at Princeton for two years, I'll introduce you to Tiger Inn. The prettiest and hottest girls in Princeton are having a blast there."
"Great-Key!"
"Great wonder!"
The otaku were thrilled; Shen Qi's image in their hearts grew even more radiant and majestic. The old professors, however, were unlikely to take them to clubs to meet pretty girls.
"Getting back to the point, the key point is that Lisa, who majored in economics, also studied basic mathematical theory, but she doesn't believe that more than two parallel lines can be drawn through a point outside a straight line." Shen Qi tapped the small dot on the blackboard and said, "What Lisa can't do, we can do for her. Euclid's Fifth Postulate 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 freehand on the blackboard, a really perfect circle, as round as this ○
Two points A and B inside a circle are connected by a curve, which represents a straight line in Non-Euclidean Geometry.
"From a cosmic perspective, we can draw countless parallel lines through a point outside of AB."
Shen Qi reviewed the concepts of Roche points and Roche lines with the students to reinforce their understanding.
Professor Muller lectured very quickly in the large lecture hall, and after two large lectures, he had already started on Riemannian Geometry.
Tutoring sessions are necessary because these second-year students are still in their academic development stage and need one-on-one individual guidance; otherwise, they may become frustrated.
The great Euclid left a flaw: the universality of his Fifth Postulate is questionable, and this is precisely the beginning of Non-Euclidean Geometry.
Non-Euclidean Geometry is divided into two main schools: Lobachevsky geometry and Riemannian Geometry.
Roche geometry, or Hyperbolic Geometry, focuses more on geometric perspectives. Riemannian Geometry, or elliptic geometry, due to the introduction of manifolds, intersects Riemannian Geometry differential calculus, topology, and Group Theory, making it more difficult to study.
In Roche's geometry, Shen Qi deduced, through graphical construction, an infinite number of parallel lines passing through a point outside a straight line. The traditional concept of parallelism disappeared, and parallelism in curved space obeyed Roche's parallel postulates.
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 rigid, traditional triangle. Shen Qi used the "devil's trumpet flower" pattern to vividly illustrate the principle that the sum of the interior angles of Roche triangle is less than 180 degrees.
Luca was a very talented student. He applied what he learned to create a set of "Great Devil Triangles," demonstrating that in Roche geometry, there are no pairs of dissimilarity similar triangles.
Italy seem to have a natural inspiration and intuition for drawing, painting, and architecture. Shen Qi admired Luca, his Italy student, Luca was talented, had a distinctive appearance, and maintained the flamboyant style typical of Italy men.
Luca isn't a homebody; he has a girlfriend. He's dated several women since coming to the US, and he's currently dating an American woman. She's quite pretty and is the daughter of the mayor of Aniston, Calhoun County Alabama.
At Princeton, it's really hard for underachievers to find a girlfriend.
Only when boys have mastered their studies can they have the energy and time to socialize, participate in sports to show off their muscles, and thus attract and meet pretty girls.
Shen Qi also Shen Qi Jack Ma, an Asian boy, and discovered that Jack Ma liked to incorporate algebraic processing into geometric analysis.
Jack Ma explained the geodesic linearity problem in constant curvature space using a set of elegant two-dimensional manifolds, which earned him praise from Shen Qi.
The academic atmosphere in the Recitation Class was quite lively. Shen Qi won the respect and goodwill of all the students with his profound mathematical knowledge and easy-to-understand explanation style.
As a gift, Shen Qi left only one homework question. He drew a very beautiful graphic on the blackboard, which looked like a reflection of the Eiffel Tower displayed together with the original image.
"I hope to receive your assignments within three days. My email address is written on the blackboard. You can send me electronic versions or submit paper versions. My office is Room A02 on the 2nd floor of the Luder Hall. Feel free to come see me anytime."
"At the start of the next Recitation Class, I will discuss this topic with you. Well, that concludes today's Recitation Class. These have been the most enjoyable two hours I've had since arriving at Princeton; you are all wonderful."
Shen Qi smiled, waved, and left the classroom.
The twelve students fell into deep thought, staring at the beautiful yet strange pattern on the blackboard. Not one of them left the classroom for half an hour. It was already 10:30 p. m., and for these second-year students, evening self-study had only just begun.
The task of a teaching assistant is to assist the professor and guide undergraduates in completing core courses.
After spending two enjoyable hours with a group of adorable young men from various countries, Shen Qi threw himself into his main task, which was, of course, scientific research.
The proof of the "Muller -Shen theorem" is proceeding as planned.
Shen Qi spent three days completing a minor detail, proving that every weakly closed convex set is a close-approach set such that ||x*-yn*||→dist(x*, A*) (n→∞) without loss of generality.
Today was the deadline for Shen Qi to collect the homework, and all twelve assignments were submitted.
The reflection of the Eiffel Tower is displayed together with the original image, making this Riemannian Geometry problem quite challenging.
Only three students successfully proved, using the Gauss -Bonnet theorem, that the Eiffel Tower and its reflection are in curved space.
The other nine students made more or less incorrect judgments and interpretations.
Copying homework is a shameful thing at Princeton, with consequences so severe that it can ruin a student's reputation.
Shen Qi believed that his boys would not copy homework, so he corrected all twelve assignments one by one.
"John, B."
"Alhilal, B."
"Luca, A."
"Stephen, A."
"Jack -Ma, A."
Shen Qi meticulously and responsibly graded the assignments, as the students' grades for the Recitation Class were kept on file.
"Jack Ma, let me take another look, hmm, A+."
Shen Qi became happier and happier as he looked at Jack Ma's homework. This kid is amazing! His handling of differentials in Riemannian Geometry is at the level I was in my sophomore year. I have to give him a plus.