Chapter 192: A One-Sided Piece of Paper

Xiao Yu silently watched as the young Luka child continued writing at a slow, steady pace.

“I heard that Lord Three and Lord Four are conducting research in this area. I saw them on TV, they’re amazing. It seems like there’s no problem too difficult for them. Only such incredible figures could become Master’s trusted aides and earn such great honors as being named Three and Four.”

“When I grow up, I’m definitely going to become a scientist too. I’ll take my mom and dad to live aboard the Paradise. I’ll have my own private lab on the Eternal Kingdom, and become a great figure like Three and Four.”

The Luka child put down its pen, and just then, the voice of the instructor came from the podium: “Alright, pens down. Time to hand in your work.”

Several Luka children stood up and, from their respective positions, gathered all the papers and delivered them to the front. After the teacher announced the end of class, the children all burst out laughing and ran out of the classroom. Xiao Yu, however, remained transfixed, continuing to stare at the scene.

In that moment, a storm surged in Xiao Yu’s mind.

He had thought of a solution, one that might actually work.

“A Möbius strip? A Möbius strip?”

Just as the Luka child had said, the method for creating a Möbius strip was incredibly simple, so simple that even a child could do it by twisting a piece of paper. But the principle contained within the Möbius strip was anything but simple.

Back on Earth, there had once been a widely circulated joke involving a Zen master. A young man asked the master why the world contained not only joy but also sorrow. In reply, the Zen master asked him whether a piece of paper could exist that had only a front side and no back.

In the joke, the young man brought out a copy of the People’s Daily, only to be immediately bested by the master.

Of course, it was just a joke and nothing more. But in truth, there was such a thing as a one-sided piece of paper, and that was the Möbius strip. All one had to do was tear a strip of paper, twist one end by 180 degrees, join it to the other, and then run a fingertip along the surface to understand why it only had one side.

In other words, the Möbius strip had no distinction between inside and outside. This was a crucial point.

Suppose we imagined a two-dimensional world as a flat sheet of paper. Drawing a circle on it would create a prison for anything existing within that two-dimensional realm. You couldn’t escape the circle from within, because there would be no openings. The circle, in two dimensions, was like a sphere in three dimensions: if you were imprisoned inside a sphere, you couldn’t escape unless you accessed a higher dimension.

In a two-dimensional world, to break out of a circle, one could achieve it through a third dimension. In the three-dimensional world, escaping a sphere could require the fourth dimension.

But the Möbius strip offered another approach. It suggested that you didn’t necessarily need a higher dimension to break confinement. In the two-dimensional world, if you could twist the plane into a Möbius strip, then since the strip had no inside or outside, you could simply walk a distance and, after restoring the space, find yourself unknowingly outside the boundary.

So if a Möbius strip could break a two-dimensional circle, was there a way to break a three-dimensional sphere?

Möbius strips only applied to two-dimensional space, and indeed, couldn’t even exist within it. They required a higher, third-dimensional space for construction.

So did there exist an equivalent of the Möbius strip that could, without using a fourth dimension, break through a three-dimensional spherical confinement?

The answer was yes: such a thing existed. It was the Klein bottle.

Imagine a bottle with no bottom. Now take its mouth and extend it downward, bending it back so that the mouth passes through the bottle’s wall and connects with what was originally the bottom. That’s a simplified version of a Klein bottle in three dimensions.

The Klein bottle was the upgraded version of the Möbius strip. The version described above was merely a projection, a simplification that could exist in three-dimensional space. A true Klein bottle, however, could not be fully constructed in three dimensions, just as a Möbius strip could not be truly constructed in two.

A real Klein bottle’s mouth didn’t need to penetrate the wall to reach the bottom. It connected through the fourth dimension, because the Klein bottle was a four-dimensional construct.

Like the Möbius strip, the Klein bottle had no inside or outside. That meant if you were inside a Klein bottle, you could simply walk forward and eventually find yourself outside, without breaching any walls. Conversely, someone outside could just as easily walk inside, unimpeded.

“Möbius strip… Klein bottle…” Xiao Yu repeated the words silently in his mind, and before he knew it, he had begun running calculations rapidly.

To create a Klein bottle, two conditions had to be met. First, one needed a profound understanding of Four-Dimensional Space. Second, one had to be able to create an actual Four-Dimensional Space.

Xiao Yu couldn’t create a Four-Dimensional Space, but there was one right beside him: the Four-Dimensional Barrier.

Which meant only two issues remained: the ability to bend three-dimensional space, a skill Xiao Yu had essentially mastered through his Faster-Than-Light Travel research and the massive energy required to do it, which he had gained through his deep study of elemental decay technology.

So what was he missing?

Nothing.

Xiao Yu began to get excited. His instincts told him that he had found the truly correct direction.

This method aligned with all of Xiao Yu’s deductions.

The Four-Dimensional Barrier really was unbreakable by Level 4 Civilizations, but the Klein bottle offered a way to bypass the barrier without damaging it.

Only Xiao Yu could utilize this method; other Level 4 Civilizations could not, because Four-Dimensional Space was inaccessible to them. And thanks to his time spent interacting with Taihao, Xiao Yu had gained some knowledge of Four-Dimensional Space, allowing him to conceptualize the Klein bottle.

“The Klein bottle might be the key to opening the Four-Dimensional Barrier… Then let’s begin the attempt.” In that moment, Xiao Yu made his decision.

He cast one last glance at the topmost test paper in the stack collected on the teacher’s podium, committing the name of the young Luka child to memory, then withdrew his consciousness.

“If I really manage to break through the Four-Dimensional Barrier using the Klein bottle, then you shall become the fourth Luka to earn a title. You shall be Luka Five.”

Xiao Yu murmured to himself, and immediately summoned all the Luka scientists to convene in the grand hall.

Every Luka scientist in attendance wore a deeply fatigued and anxious expression. Clearly, none of them had come up with any new ideas during this time, and they hadn’t rested well either.

“Master, may I ask why you’ve called us here?” Luka Three inquired.

“I have a hypothesis,” Xiao Yu replied. “I may have found a way to break through the Four-Dimensional Barrier.”

“Oh? What method? What is it?” Luka Three asked urgently.

“Klein bottle,” Xiao Yu said, uttering a single concise term.

For a moment, silence fell over the grand hall. It was as if the words Klein bottle held a mystical power, stunning every Luka present into awestruck silence.

“Klein bottle? A Klein bottle? That self-contradictory object, can it really help us?” After a long pause, Luka Three finally managed to speak with difficulty.

Indeed, the Klein bottle was riddled with contradictions. The main contradiction lay in the fact that it couldn’t truly be constructed within three-dimensional space. But it was precisely those contradictions that had presented Xiao Yu with a method to bypass confinement in three-dimensional space.

“Don’t forget, if a Möbius strip can allow one to break out of a two-dimensional circle, why couldn’t a Klein bottle allow one to break out of a three-dimensional sphere?”

Xiao Yu responded.

Silence fell once more in the hall. Clearly, all the Luka scientists were intensely thinking it over.

“From a theoretical perspective… Yes, I mean, purely from theory, it’s entirely feasible to use a Klein bottle to bypass the Four-Dimensional Barrier. But the question is, how do we actually create such a self-contradictory structure? After all, we are in a three-dimensional world,” said Luka Four.

“We have elemental decay technology to provide energy. We have Faster-Than-Light Travel technology to bend space. And we have the Four-Dimensional Barrier itself, which can serve as the fourth dimension to allow the Klein bottle’s mouth to connect to the base without passing through the surface. Why can’t we build one?” Xiao Yu countered.

“Well… that does sound logical,” Luka Three said hesitantly. “In any case, this idea is worth testing.”

“Of course,” Xiao Yu said. “It’s our last remaining option. If it works, we break through the Four-Dimensional Barrier. If it doesn’t, then we have no other choices left but to sit and wait for death. So, let’s begin the effort. I’ll need your help to compute the curvatures and figure out how to create a Klein bottle…”