Riemannian geometry, on Earth, was proposed by the German mathematician Riemann at the University of Göttingen. This is an extension of Euclid's fifth postulate, an expansion of Euclidean plane geometry.
Due to a mysterious correlation between the two worlds, or perhaps merely a perverse amusement of some Heavenly Dao, this geometry with positive curvature is also referred to as "Riemannian Geometry" or "Li's Geometry" in Shenzhou. The Carefree Practitioners of Wanfa Sect and the Surface Sky Demon Riemann achieved enlightenment through this geometry.
Riemannian geometry is the mathematical foundation of general relativity, used to describe the spacetime model of general relativity. "The essence of gravity is the curvature of spacetime," and this "curvature" refers to the curvature in Riemannian geometry.
Thus, Ai Changyuan also incorporated Riemannian geometry into the Sky Collapsing Palm, enabling him to master this miraculous technique during the Qi Cultivation Stage.
Originally, Wang Qi wasn't very confident about defeating Ai Changyuan, but after witnessing yesterday's match, he spotted an opportunity for victory.
"You use Riemannian geometry, integrating it into the Sky Collapsing Palm, which reduces the difficulty of this martial arts and simultaneously enhances its power, which is good—but it also means you have to rely on generating pellets for your moves!"
Wanxiang Heavenly Attraction evolved from the true form of gravity, formless and featureless, extremely hard to deal with. Not to mention its ability to directly interfere with blood flow can't be underestimated. As an adjective, "Heart Blood Comes" implies high spirits, but as a verb or noun without that connotative edge, it can turn lethal.
Since Wanxiang Heavenly Attraction doesn't have a Qi Cultivation part, the Immortal Institute Transmission Hall doesn't have it recorded. Being a descendant of the Yuanli Master, the Ai Family from Split Peak can learn this strange skill. Without knowing the mental method, it's difficult for Wang Qi to spot any flaws.
However, the traces of Riemannian geometry are detectable! Especially for someone who excelled at it in a previous life.
"So, what I need to do now, is integrate Riemannian geometry into my saber technique—not to create a whole new set of saber techniques rooted solely in Riemannian geometry, but to blend its mathematical reasoning with my saber skills just enough!"
What Wang Qi needs to do now is not to create a new sword technique, but to add a touch of Riemannian geometry's reasoning to the Bai Ze True Formation Slash and Thunder Sound Chord Saber Technique.
Riemannian geometry closely relates to differential manifolds, partial differential equations, and multiple complex functions. The Thunder Sound Chord Saber Technique and Bai Ze True Formation Slash also originate from calculus. Wang Qi hadn't considered this path earlier simply because he hadn't integrated the most crucial concepts of calculus like "limit" and "infinity" into his martial arts.
The more fundamental the concepts, the harder they are to be applied proficiently. Originally, Wang Qi would have needed months to years to reach this point, but an accident a few days ago saved him the labor.
The Spirit Treasure Fractal Map, a mysterious diagram ingrained within Wang Qi's soul, also contains the mystery of "infinity." During the three days he calmed his Heart Demon, he also slightly touched upon some mysteries of the fractal map, which helped him find a way to integrate calculus into his saber technique.
Of course, it's just the initial path so far. Wang Qi is still a long way from mastering it. Yet even so, compared to an ordinary person who hasn't found this path, he is far ahead.
While Wang Qi painstakingly calculated a set of saber techniques to counter the Sky Collapsing Palm, Ai Changyuan was doing something similar. He pondered over the Breaking Technique for the Elephant Phase Wave Skill and Heavenly Entropy Technique.
This young man was diligently researching in the study room. Unlike Wang Qi, Ai Changyuan, despite his pride, always put forth his best effort like a lion fighting a rabbit—an even more commendable trait was that he never courted disaster. On this front, a certain disaster-courting fool really should learn from him.
The Cultivators of the Yuanli Master's era had a strong commitment to causality. The Ai Family from Split Peak retained that era's customs. Growing up with this constant reinforcement, Ai Changyuan firmly believed a saying by the Bai Ze God Monarch—"The outcome of a battle between Cultivators is decided before the fight even starts."
After the Mysterious Palace established the Mysterious Dao, determinism was somewhat shaken, and many Cultivators began debating meaningless questions like "What if a meteor suddenly flies over while I am fully prepared and excited, and kills both me and my opponent?" coming to the conclusion that "the result of a battle is probabilistic." However, Ai Changyuan never believed this. He always prepared as thoroughly as possible.
It had just passed the winter solstice, the days were short, and night soon fell. Lu Qianqian, who was with Ai Changyuan, activated the illumination Magic Seal in the room. Seeing her friend's furrowed brow, she was a bit worried and asked, "Changyuan, Changyuan, is this opponent particularly troublesome?"
Ai Changyuan nodded. "Indeed. I'm struggling to find a foolproof method to defeat him. My only strategy right now is to avoid close combat as much as possible."
Lu Qianqian was greatly surprised. "Why? Changyuan, your hand-to-hand combat skills are so strong!"
"The issue isn't with the combat skills, but with the mental method." Ai Changyuan put down the pen in his hand and asked, "Do you remember his first match?"
Lu Qianqian quickly nodded. "Yes, Kun Zhongxuan!"
"All he did was lightly guide with his fist, and Kun Zhongxuan's 'Earth Qi Gang' automatically collapsed from within. That's the malevolence of the Heavenly Entropy Technique. If the power of Entropy Increase invades the body, at best it causes a momentary stagnation in Qi, at worst it leads to Deviation and death—with spontaneous combustion being the most common way to die." Ai Changyuan explained, "I'm strong in attack but weak in defense. If my internal energy gets disrupted, it would spell big trouble."
Lu Qianqian blinked. "So, use Wanxiang Heavenly Attraction from a distance to control the opponent's blood?"
"I've never faced a Practitioner of the Elephant Phase Wave Skill before, so I don't know the mysteries of this technique. However, since it involves Phase Waves, it likely has the capability to disturb gravity."
"Are you thinking of using Sky Collapsing Palm right from the start?"
"I'm considering it, but I still need to think it over," Ai Changyuan closed his eyes and nodded, "and I can't ignore his mastery in saber techniques."
The same guy who is also strong in attack but weak in defense like me, is it possible to take him down with a round of strong attacks… No, 'weak defense' is relatively speaking, and if he surrounds himself with Phase Wave Protection... I have yet to see someone break through that defense...
Watching Ai Changyuan deep in thought, Lu Qianqian suddenly felt full of confidence: "No matter what, Changyuan, you're the best! You can definitely win tomorrow!"
Ai Changyuan smiled and patted the girl's head, his thoughts drifting elsewhere.
Tomorrow... you won't disappoint me, will you?