A scholar of Zhenguan in the Tang Dynasty

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"I have a question, follow me." Gu Qing took Gu Xiao to his laboratory, closed the door and locked it, opened the secret room, and took out a glass bottle from the secret room.

"Isn't this just a bottle?" Gu Xiao was surprised.

"This bottle is the problem." Gu Qing said after closing the door of the secret room, "It is said that there is a bottle that can never be filled with water. This is a puzzle in the world. It is also called the Klein bottle. In fact, the origin of this bottle is Story of."

"This bottle is used to hold water." Gu Xiao looked curiously.

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Chapter 9 The Disappearing Side

"Actually, it's not used to study how much water it can hold, it's used to study planes." Gu Qing continued, "We all know that things have both sides, a piece of paper has a front and a back, but this Klein The bottle has only one side, so it was originally used to study noodles."

"I don't understand." Gu Xiao shook his head vigorously.

After Gu Qing tore a piece of paper and twisted it into a ring, he said to Gu Xiao, "Now look at how many sides this paper has."

Gu Xiao looked at it for a while and found that his expression was very confused. No matter how he followed this distorted trajectory, he would come to the original point. "This seems to have two sides, it seems to have only one side, so amazing."

"This is called a Mobius ring." Gu Qing then made a Mobius ring again, and after connecting the two in parallel, one looks like a Clay bottle. Then he said, "The Mobius ring is for research , and this Klein bottle is used to demonstrate the Mobius ring. In fact, the Klein bottle in my hand is wrong. It can be filled with water, because the real Klein bottle is in this It cannot be made in the world, if it really exists, it only exists in the four-dimensional space."

"What is a four-dimensional space." Gu Xiao then asked.

"The four-dimensional space is a space beyond our own, and our space is a three-dimensional space." Gu Qing continued, spreading out a piece of paper and said, "This paper is a two-dimensional plane, and we can easily draw a picture on it. Circle, because we are in three-dimensional space, if we are in four-dimensional space, killing a person is as simple as drawing a circle on a two-dimensional plane in three-dimensional space."

"A long time ago, there was a woman. Many people thought she could use magic, because she could take the heart of a person in another place. Many people thought it was miraculous. After that, this woman's ability could no longer be used. Then It is because she briefly came into contact with a fragment of a four-dimensional space, just as it is so easy for us to draw a circle on a two-dimensional plane in three-dimensional space, she can also do it with ease." Gu Qing looked at her son and said, " Take it as a story. The existence of the four-dimensional space has always been a mystery, and it is also a mystery left by human beings for thousands of years. No one can solve this mystery. Perhaps the vast universe is what they call the possibility There is an answer."

"And this Möbius ring is an argument for ascending from two dimensions to three dimensions." Gu Qing continued, "And this Klein bottle is an argument for demonstrating the ascension from three dimensions to four dimensions, but we live in three dimensions. , in our hands it is impossible to make a bottle that is not filled with water."

"Oh." Gu Xiao nodded vigorously, and a puzzle was in his mind at this moment.

Gu Qing said again, "The puzzle you gave to Li Chunfeng is, when a piece of paper with two sides becomes a Möbius ring, why does it only have one side, where does the disappearing side go, and this bottle, what is this bottle? One side or three sides or four sides."

Gu Xiao was very interested in holding the Mobius ring. After repeated experiments for a while, it was true that he would return to the original point no matter where he started. I don't know the relationship between [-]D, [-]D and [-]D, I'm afraid Li Chunfeng will go crazy

"You." Gu Qing poked Gu Xiao's forehead, "Stop tossing around in the academy, and listen to the masters, Li Chunfeng, don't get too close, this old guy also likes to study some weird things, you Do not follow the research."

"My son understands, my son was not interested in those things." Gu Xiao said holding the Klein bottle.

Walking out of the laboratory, Gu Xiao was still confused. After eating, he came to Gu Qing's side and asked, "Father, what would it be like if we could enter the four-dimensional space."

"Four-dimensional space?" Gu Qing looked up at the starry sky, "We can't enter it. If we enter the four-dimensional space, we are transcending time. Time is a dimension in the four-dimensional space."

"Oh." Gu Xiao also looked up at the starry sky.

Early the next morning, when Gu Xiao returned to the academy, Zhang Jiuling saw Gu Xiao coming back at the gate of the academy and hurried forward, "Gu Xiao, have you found the topic yet?"

"I found it." Gu Xiao said confidently holding the Klein bottle.

"Why don't you forget it, just apologize to Mrs. Li Chunfeng." Zhang Jiuling said, "Master Li Chunfeng is a master of numerology in the academy, and your question will not trouble him."

"Do you know who my father is?" Gu Xiao said confidently, "Brother Zhang! I heard Grandpa Shigong said that Master Li Chunfeng still asked my father for advice when he was young? Don't worry, I won't lose. "

"I know your father is Gu Qing." Zhang Jiuling was in a cold sweat.

"I will apologize if I apologize, but apology and competition are two different things. I will not lose to Li Chunfeng." Gu Xiao strode into the academy and all the way to Li Chunfeng's office.

Zhang Jiuling followed all the way here and hurriedly saluted Li Chunfeng, "I've seen Daoist Li Chunfeng."

"Little devil, have you found the problem?" Li Chunfeng looked at Gu Xiao.

"That's right, I found the problem." Gu Xiao said, "Master Li Chunfeng, it's still too late for you to admit defeat."

"Admit defeat? Heh!" Li Chunfeng sneered, "Let's come up with a question."

"Okay!" Gu Xiao took a piece of paper and said, "Master Li Chunfeng is optimistic, this is a piece of paper, right! This piece of paper has two sides, right?"

"Yes." Li Chunfeng nodded.

"Master Li Chunfeng, believe it or not, I can make the other side of this paper disappear."

Listening to Gu Xiao's words, Li Chunfeng sneered, "How is it possible, unless the poor man is blind."

"Then you can take care of it." Gu Xiao twisted the paper in his hand according to the method taught by Baoqing yesterday, and spliced ​​it end to end. After the glue is completed, a Mobius ring is ready. "Okay, now Li Chunfeng Master, look at this circle made of paper, it has several sides."

Li Chunfeng curiously took the ring made by Gu Xiao and said, "Even if you connect the first part of the paper, there are two sides, and it is impossible for one side to disappear."

"If it's on two sides, the two points will not collide if they are separated from each other. You should take a good look." Gu Xiao sat in front of Li Chunfeng like an old man.

Li Chunfeng observed the Mobius ring in his hand. After several observations, he found a clue. He glanced at Gu Xiao in surprise, then took out the brush and started to move along the Mobius ring until it returned to the original point. Li Chunfeng was shocked. I couldn't hold the brush in my hand.

Zhang Jiuling was also shocked. The ball has only one side, and it can return to the original point as long as it follows one direction, because the ball has only one side, but Gu Xiao's ring very cleverly twisted the two sides of the paper to become one side.

Chapter 961 Academy Regulations

"Is this a trick?" Li Chunfeng was surprised for a long time, as if breaking a conventional concept.

Zhang Jiuling is also very curious if the concept of throwing away both sides is actually not true, it is a kind of distortion and overlap.

As if opening a new door, Li Chunfeng observed the Mobius ring in his hand, which seemed to be a kind of knowledge that he had never come into contact with. If it was geometry, he had mastered it very proficiently, but the Mobius ring in front of him Rings seem to be related to geometry.

But this situation is calculated geometrically, the area of ​​the two sides of a piece of paper remains the same, it is just summed up, but it is indeed the same as Gu Xiao said, there is only one side.

"The old man lost." Li Chunfeng looked at Gu Xiao, "But the old man didn't lose to you, he lost to your father. This question was written by your father, right?"

"That's right." Gu Xiao didn't deny it either.

"Stinky boy! You have the same virtue as your father." Li Chunfeng said with a smile, "I got a new question so that the students of the academy can study it. Specialize in your topic. What is it called?"

"Mobius Ring." Gu Xiao said.

"Why is there such an inexplicable name, it's a mouthful." Li Chunfeng looked aside and saw the glass bottle and asked curiously, "What is this bottle?"

"This bottle is also my father's subject." Gu Xiao put the bottle in front of Li Chunfeng.

Li Chunfeng picked up the strangely shaped glazed bottle, "What kind of question is this again?"

"Compare this Mobius ring with a bottle." Gu Xiao said with a smile.

Zhang Jiuling stood aside and listened to the conversation between Gu Xiao and Li Chunfeng. At the same age, this Gu Xiao was two years younger than himself, but his conversation was much more natural than his own. Perhaps it was a relationship of superiority and inferiority. Gu Xiao was born It determines how tall he is, Gu Xiao is almost a little master in the academy.

Knowing that Gu Xiao had entered Li Chunfeng's office as soon as he came to the academy, Li Linfu and Yang Zhao hurried over to see Fang Chengxian standing outside the door and hurriedly asked, "What's the matter?"

"Keep your voice down." Fang Chengxian said hastily.

"Did Gu Xiao get into trouble again?" Li Linfu asked.