Chapter 90: Geometry of the Light Wings and Hodge Conjecture

After spending a few more years together, Xin developed a liking for Yan Xia, the "kind" leader, and gradually opened his heart to talk and even debate with Yan Xia.

"Mathematics is a powerful tool that gives scholars the imagination to understand the universe."

"According to the information I have obtained, there is a five-hundred-year-long mathematical craze in my civilization, and we have a strong pursuit of excellent results in the field of geometry."

Xin was lying on an iron ball as big as a football. The ball would release weak static electricity, and Xin seemed to like the feeling of being stimulated by static electricity very much.

They were staying in the library, and the reason for talking about this was because Yan Xia was studying geometry.

Geometry is a very important classification in mathematics, and its most important application is in spatial structure. If you want to understand the concept of multidimensionality, geometry is almost a must.

Now the Civilization Federation is digging into multidimensional space, and the mainstream academics are line theory, superstring theory and M theory.

Many scholars who explore this theory also study geometric mathematics and have great success.

Xin also spoke up when he saw this.

The information in this library was moved from the capital galaxy. Although the books are made of paper, the ink used is very special and can release weak heat, allowing creatures without eyes that can sense heat to see the text on the books.

Although Xin does not understand Chinese characters, some of the mathematical figures on it can be understood through simple calculations.

"Even so, our civilization still has not achieved unity or decomposition in the field of geometry."

"If it can be reached, then our civilization will not be helpless in front of unknown civilizations."

Yan Xia put the book down and was very interested in the new topic.

Or he actually likes to listen to stories.

Just like the old man can chat all day.

"Geometry corresponds to the understanding and imagination of space. The famous three-dimensional geometric structure is the Möbius strip."

Yan Xia stretched out his hand to summon a holographic panel, then pulled up a rectangle, fixed one end of it, twisted the other end 180 degrees, and finally connected the two ends together, and a Möbius strip was formed.

This is a very representative non-Euclidean geometric body. Interestingly, the Möbius strip is very similar to the symbol of infinity ∞.

The reason why the Möbius strip is famous is that it can be made in the real dimension and has magical mathematical properties.

I was very excited when I saw the Möbius strip for the first time.

"Yes, that's it."

"Any figure in the universe can be called a geometric figure, and points, lines, and surfaces in space are also part of geometry."

"The Carletto infinite strip, which is the Möbius strip of your civilization, is one of the basic geometric structures of our civilization in addition to various squares, spheres, and circles. A lot of technology in our civilization uses this geometric body."

Yan Xia remembered seeing the geometric structure of the Möbius strip in those spacecraft wreckages. In fact, it was for this reason that he studied geometry in recent decades.

Xin continued: "The Möbius strip has a characteristic, that is, after cutting it horizontally, it will form two Möbius strips nested together, and if it is cut along these two Möbius strips, it will form four Möbius strips nested together, and so on."

"From here, a great mathematician of our civilization thought about a question, how to split the Möbius strip?"

"The splitting here is not simply cutting, but destroying the geometric structure of the Möbius strip and turning it into a more understandable geometric structure, such as a square, rectangle, etc."

"Unfortunately, he never split the Möbius strip in his life."

This is the "string theory" or "grand unification" in the field of geometry.

Any complex structures can be disassembled into simpler geometric structures.

The moment Yan Xia heard this sentence, he thought of the Hodge conjecture, one of the seven famous mathematical problems of the 21st century.

That is: "Every harmonic differential form on a non-singular projective algebraic variety is a rational combination of the cohomology class of an algebraic closed chain."

This is a very complex mathematical problem, which includes concepts such as geometry, algebra, and calculus.

Yan Xia knew that the Hodge conjecture was not seen in a book, but one of the biggest research directions in the field of mathematics in the Civilization Federation at this time was the Hodge conjecture.

Because the Hodge conjecture, Fermat's Last Theorem, and Riemann's conjecture are the carriers and tools of the M-theory structure geometry and topology that integrates general relativity and quantum mechanics.

As the mainstream theory of the Civilization Federation now, M-theory requires the operation of the Hodge conjecture on spatial geometry.

This complex conjecture can actually be roughly and simply understood as any geometric figure is a house, which is built of bricks and tiles, which means it can be split.

It is actually very easy to understand in the low-dimensional field.

It is well known that the movement of points forms lines, and the movement of lines forms surfaces. The same is true in reverse. Surfaces can be split into lines, and lines can be split into points.

It's just that the Hodge conjecture is not such a simple analogy, but a deeper and rougher algebraic and geometric combination problem.

Currently, Hodge's conjecture has not yielded a substantial answer. If it could, then the Civilization Federation would have begun to touch upon the true mystery of dimensions, a realm that even Level 2 civilizations dare not imagine.

It could even be possible to cross dimensions, and thus create cross-dimensional weapons, such as - two-dimensional foil.

It is indeed as Xin said. If the Light Wing Clan civilization can make a breakthrough in this area, it will not be helpless in front of unknown civilizations.

"In fact, our civilization is not without any gains."

"During hundreds of years of exploration, we gradually disassembled the Möbius strip into simpler geometric figures, a rectangle and a twisted rectangle."

"Or it can be a rectangle twisted once and a rectangle twisted twice."

"These decompositions allow us to create alternative energy closed-loop applications."

"Imagine a beam of light bombarding the Möbius strip, allowing it to enter the inner space from the surface space, and split the Möbius strip before the light escapes from the inner layer, so that the photons turn the Möbius strip into a ring before completing the inner path of any geometric figure in the split rectangle or twisted rectangle, locking the energy in it."

"However, due to the limited technological level of our civilization, we can only make nano-level geometric closed-loop transformations in the microscopic field, and this tiny application cannot have much impact on the macroscopic field."

The Light Wing Clan's exploration in the field of mathematics has obviously reached a height that is inconsistent with the level of civilization.

Just like a longer piece of wood on a wooden barrel.

Xin said this for a purpose, which was to come up with something to prevent Yan Xia from going back on his word.

Yan Xia speculated that this was related to his interaction with Ella yesterday.

Ella's attachment needed his permission to do many things, which made Yan Xia a little impatient, so sometimes Yan Xia would do something inconsistent to fool Ella.

Perhaps because of this, Yan Xia's image in Xin's mind has changed a little.