Chapter 90: The Geometry of the Light Wing Clan and the Hodge Conjecture

After a few years of getting along, the Light Wing developed a fondness for Yan Xia, the "kind" leader, and gradually opened up to him, even engaging in debates.

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

"According to the information I've received, my civilization went through a fifty-year period of mathematical fervor, and we had a strong pursuit of excellence in geometry."

The Light Wing was lying on a soccer-sized iron ball that emitted a weak static charge, which it seemed to enjoy.

They were in the library, and the Light Wing brought up the topic because Yan Xia was studying geometry.

Geometry is a crucial branch of mathematics, primarily applied to understanding spatial structures. To comprehend the concept of multiple dimensions, geometry is essential.

The Galactic Federation was currently exploring multi-dimensional space, with string theory, superstring theory, and M-theory as the leading academic fields.

Many scholars studying these theories also learned geometry and achieved significant success.

The Light Wing had noticed this and spoke up.

The library's collection was brought from the capital star system, and although the books were made of paper, the ink was special, emitting a faint heat that allowed beings without eyes but with heat-sensing abilities to read the text.

The Light Wing didn't understand Chinese characters, but it could understand some of the mathematical diagrams through simple calculations.

"Despite this, our civilization never achieved unity or a solution in the field of geometry."

"If we could have reached that level, our civilization wouldn't have been so helpless in the face of an unknown civilization."

Yan Xia put down the book, genuinely interested in the Light Wing's topic.

Or perhaps he just enjoyed listening to stories.

Like an old man who could ramble all day.

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

Yan Xia summoned a holographic panel and created a rectangle, fixed one end, twisted the other end by 180 degrees, and then connected the two ends, forming a Möbius strip.

This is a remarkable non-Euclidean geometric structure, and interestingly, the Möbius strip closely resembles the infinity symbol, ∞.

The Möbius strip is famous because it can be created in real dimensions and has fascinating mathematical properties.

The Light Wing was thrilled to see the Möbius strip.

"Exactly, that's it!"

"Any shape in the universe can be called a geometric shape, and points, lines, and planes in space are all part of geometry."

"The Calabi-Yau manifold, or your civilization's Möbius strip, is one of the basic geometric structures in our civilization, and many of our technologies utilize this geometric shape."

Yan Xia remembered seeing the Möbius strip geometry in the wreckage of the ships, which was why he had been studying geometry for the past few decades.

The Light Wing continued, "The Möbius strip has a unique property: when you cut it horizontally, it forms two interlocked Möbius strips. If you cut these two strips, they become four interlocked strips, and so on."

"Based on this, one of our civilization's great mathematicians pondered a question: how can we disassemble the Möbius strip?"

"By disassembling, I don't mean simply cutting it, but rather breaking down the geometric structure of the Möbius strip into simpler geometric shapes, such as squares or rectangles."

"Unfortunately, he spent his entire life trying to disassemble the Möbius strip without success."

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

Any complex structure can be broken down into simpler geometric shapes.

When Yan Xia heard this, he immediately thought of the Hodge Conjecture, one of the seven most challenging mathematical problems of the 21st century:

"Every harmonic differential form on a projective algebraic manifold is a rational linear combination of closed, algebraic differential forms of the same degree."

This is a highly complex mathematical problem involving geometry, algebraic geometry, and calculus.

Yan Xia knew that the Hodge Conjecture wasn't just something he had read about; it was one of the primary research directions in the Galactic Federation's mathematical field.

The Hodge Conjecture is the carrier and tool for the geometric topology of M-theory, the mainstream theory of the Galactic Federation, which combines general relativity and quantum mechanics.

M-theory, the current mainstream theory of the Galactic Federation, requires the Hodge Conjecture for spatial geometric calculations.

This complex conjecture can be roughly and simply understood as follows: any geometric shape is like a house built with bricks, implying that it can be disassembled.

Understanding disassembly in lower dimensions is relatively easy.

It's common knowledge that the movement of points forms lines, the movement of lines forms planes, and vice versa.

However, the Hodge Conjecture is not a simple analogy but a deeper problem involving algebraic geometry.

Currently, the Hodge Conjecture has not yielded substantial results. If it could be solved, the Galactic Federation would begin to touch upon the true mysteries of dimensions, a realm that even Level 2 civilizations dare not imagine.

It might even be possible to create trans-dimensional weapons, such as—Metamaterials.

Indeed, as the Light Wing said, if they had made breakthroughs in this area, their civilization might not have been so helpless against the unknown civilization.

"In fact, our civilization did make some progress."

"After fifty years of exploration, we gradually disassembled the Möbius strip into simpler geometric shapes: a rectangle and a twisted rectangle."

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

"These solutions allowed us to create alternative energy loops."

"Imagine shooting a beam of light at a Möbius strip, causing it to enter the inner space from the outer layer. Before the light can escape from the inner layer, we disassemble the Möbius strip into a rectangle or a twisted rectangle, or any other geometric shape, and then turn the Möbius strip into a circular ring, trapping the energy within."

"However, due to the limitations of our civilization's technology, we could only create nanometer-scale geometric loops, which had little impact on the macroscopic world."

The Light Wing's exploration of mathematics, especially geometry, had reached a level beyond what their civilization's level could support.

It was like the longest plank in a barrel.

The Light Wing had a purpose in sharing this information—to offer something to prevent Yan Xia from going back on his word.

Yan Xia suspected that this was related to an incident between Ella and him the previous day.

Ella's attachment often needed his permission to perform tasks, which annoyed Yan Xia, so sometimes he would change his mind to tease Ella.

Perhaps this had caused a change in Yan Xia's image in the Light Wing's perception.