"Dean, teachers, let me start with the Klein bottle. Because in the future, it may be necessary to gather the strength of the people to do one thing. Qin Fengxian hugged the qin, bowed to everyone, bowed slightly, and said, "First of all, the translation of the name Klein Bottle, in fact, is somewhat wrong. The original name came from a country where the galactic civilization originated, Germany. When named in German, it's called 'Kleinsche fl?? che' is the meaning of 'Klein plane'. Because it was written as Flasche when it was translated, it became the meaning of the bottle used in the current term, and it was called the Klein bottle by later generations. But don't worry, the word 'bottle' is a good fit. ”

"The Klein bottle was discovered by the famous German mathematician Felix Klein in 1882. This is a surface, closed like a sphere, that is, without edges. But there is only one side to it. Above the Academy, we see that the Klein bottle is indeed like a bottle. But it doesn't have a bottom, its neck is elongated, then it seems to pass through the wall, and finally the neck is connected to the bottom circle. If the neck of the bottle does not pass through the wall and is connected to the bottom ring from the other side, we get a torus. ”

"The Klein bottle is a non-orientable two-dimensional compact manifold, while the sphere and millstone interview can be directed two-dimensional tight manifolds. If you look closely at the Klein bottle, one thing seems confusing: the neck of the Klein bottle and the bottle body intersect, in other words, some points on the neck and some points on the wall of the bottle occupy the same position in three-dimensional space. ”

"When we put the Klein bottle in a four-dimensional space, understanding is perfect. The Klein bottle is a surface that is only possible in a four-dimensional space. If we have to express it in the three-dimensional space of our lives, we have to settle for it and act it as if we are intersecting with ourselves. The neck of the Klein bottle passes through the fourth dimension and is connected to the bottom circle of the bottle, and does not pass through the wall of the bottle. To use the kink analogy, if you think of it as a curve on a plane, then it seems to intersect itself, and then it seems to break into three. But it's easy to understand that this figure is actually a curve in three-dimensional space. It does not intersect with itself, but is a continuous curve. A curve can't do this on a plane, but if it has a third dimension, it can pass through it to avoid intersecting itself. ”

"It's just that when we want to draw it on a two-dimensional plane, we have to draw it as if it's intersecting or breaking. The same goes for the Klein bottle, which we can think of as a surface in a four-dimensional space. In our three-dimensional space, even the most skilled craftsmen have to make it intersect with themselves. It's like the most brilliant painter who has to paint knots on paper as if they intersect themselves. Interestingly, if you cut the Klein bottle along its line of symmetry, you get two Möbius rings. ”

If the Möbius strip can perfectly represent a "one-dimensional infinitely expandable space model in two-dimensional space", the Klein bottle can only be used as a reference to show a "two-dimensional infinitely expandable space model in three-dimensional space". Because in the process of making Möbius tapes, we need to flip the paper tapes 180° and then connect them end to end, which is an operation in a three-dimensional space. The ideal "two-dimensional infinitely expandable space model in three-dimensional space" should be a model that can return to the origin in any direction in a two-dimensional plane, while the Klein bottle can move infinitely in any direction on a two-dimensional plane. ”

But only in two specific directions will it return to the origin, and only in one of the directions, before returning to the origin, will it pass through a "reverse origin", and the real ideal "two-dimensional infinitely expandable space model in three-dimensional space" should also go in any direction on the two-dimensional plane, and will first pass through a "reverse origin", and then return to the origin. To make this model, the three-dimensional model needs to be distorted in four-dimensional space. ”

"Klein flasks are defined as square regions [0,1]× [0,1] which mold off equivalence relations (0,y)~(1,y), 0≤y≤1 and (x,0)~(1-x,1), 0≤x≤1. Similar to the Mobius Band, Klein bottles are not orientable. But the Mobius strip is embeddable?? , while the Klein bottle can only be embedded in a four-dimensional (or higher-dimensional) space......"

When Qin Fengxian finished telling the Klein bottle and took out the paper and pen to sketch it, except for Song Xiaopeng, everyone else watched and listened in a daze.

They didn't understand what was the point of Qin Fengxian's explanation?

As everyone knows, Song Xiaopeng listened very seriously.

The sub-saint-class destroyer and starship are made of these principles and refined with the so-called 'super technology'.

Super technology, a bit similar to Song Xiaopeng's carving.

What kind of topology is the Möbius belt in Qin Fengxian's mouth, even though there is the Tree of Civilization as an aid, Song Xiaopeng actually doesn't understand it very well, but he understands the meaning.

"Teacher Fengxian, with the current conditions in the academy, can we have a chance to research a sub-saint-class destroyer?" Song Xiaopeng asked.

"If the strength of the Phoenix Immortal can be raised to a higher level, in five to ten years, the sub-saint-class destroyer can be researched. "Since Qin Fengxian was promoted to the Qingye realm, her face has recovered a lot, and compared to the previous gentle and elegant, she has added more indescribable temperament, giving people an inexplicable convincing.

"Okay, in addition to the nine faculties, there is a super science and technology department, and Qin Fengxian will be in charge, and I will work hard for Teacher Fengxian in the future. Song Xiaopeng made a decision on the spot, this kind of personnel order, he can completely call the shots.

"Impatiens should be. Qin Fengxian nodded.

"Public and private together, just weigh the balance. Song Xiaopeng also knows that Qin Fengxian has her own selfishness, but it doesn't matter, what selfishness Qin Fengxian has, it will not affect the development of the academy, on the contrary, she may use the power of selfishness to make more and greater contributions to the academy, why not?

On the contrary, looking at those people who were still listening to the book of heaven, Song Xiaopeng smiled and said: "Dark web civilization, dark civilization, ancient Greek civilization, presumably you are all strangers, more or less have you dealt with each other, right?"

"Dean......" Everyone was stunned.

"Then do you know how they came here?with strength?" Song Xiaopeng smiled and said sonorously, "Naturally not, their strength wants to cross the galaxy to come here, it is tantamount to a fool's dream." It is they who have the Lesser Saint-class destroyers who can cross the distant galaxy and descend to our homeland. ”

"Then we can go and grab it!" Swallow Ergou said suddenly.

"Can you grab this thing?" but you ask the coconut pear, can you easily grab it?" Song Xiaopeng looked at the coconut pear as he spoke.

"Can't grab it, first of all, the sub-saint-class destroyer has the unique mark of our clan on it, and even if someone can forcibly erase the imprint, it can't be used, and we need a specific energy ......" Coconut pear has the most right to speak, at this moment, under Song Xiaopeng's signal, the restrictions of the sub-saint-class destroyer are eloquent, "Even if someone can hunt my aunt, the special energy in her hand will dissipate with her, unless she is willing to use it, someone can move the sub-saint-class destroyer." ”

Coconut pear's words can be said to dispel Song Xiaopeng's thoughts, including the minds of the people in Xinghe Academy.

"What Mr. Coconut Pear said is good, however, there are exceptions, when you meet a saint, or an alchemy master, even if you don't have special energy, you can crack the energy groove, reshape the new energy, and drive the sub-saint-class destroyer. Qin Fengxian said lightly.

was originally silent, but Song Xiaopeng instantly became active again.

Qin Fengxian's words were undoubtedly not implying that she was an alchemy master herself.

It's just that the current strength has not yet been reached, and he can't be called a real alchemy master.

But, enough.

The Dark Clan, that is, the sub-saint-class destroyers of the Light Dream and Coconut Pear clans, he didn't have any ideas.

The two Lesser Saint-class destroyers of the Dark Web Civilization and the Ancient Greek Civilization ......

You can give it a try and grab it.

Five years and ten years is too long.

I can't wait that long.

At that time, if there is really a wormhole in the black hole whirlpool, that is, the black hole channel, he is forced to leave, and there is no sub-saint-class destroyer by his side, and the danger of entering it is incalculable. He didn't think he could stand up to the coercion in the wormhole on his own.