Gaode opened his eyes, subconsciously rubbing his temples.
Constructing a Magic Model was extremely mentally exhausting.
His apprentice-level Meditation Technique had already reached the stage of five petals. This level of Spiritual Power was enough for him to complete the model construction of a Trick, but it was still a bit strenuous. If he could cultivate it to the Perfection of sixteen petals, his Spiritual Power would be formidable enough to construct the Magic Model of a 0th Ring Magic with much greater ease.
The difficulties of constructing a Magic Model were twofold. First, it required extreme precision—the kind where a hair's breadth of error would lead to complete failure. Second, it required the Mage to possess enough Spiritual Power to sustain the exhaustion of repeated attempts.
With Gaode's current Spiritual Power, every failed attempt at constructing the Magic Model for Acid Splash left his head throbbing with exhaustion. At most, after three failures, his brain would begin to ache. The excessive depletion of Spiritual Power meant he would need to rest and wait for his reserves to recover before he could attempt another construction.
This was the drawback of having insufficient Spiritual Power. If a First-Ring Mage were to construct a 0th Ring Magic Model, their efficiency would not only be dozens of times higher than Gaode's, but even if they failed, doing so dozens of times a day wouldn't be a problem.
"Constructing a Magic Model really isn't simple. No wonder my predecessor spent over a year just to master the two Tricks, Repair Skill and Mage's Hand," Gaode murmured to himself.
If mastering a simple 0th Ring Magic was this arduous, one could only imagine how much effort was required to become a powerful Mage. However, he didn't complain.
Everyone said Mages were the true masters. But how could anyone become a master without first enduring the grind of a lowly apprentice?
...
Failure is the mother of success.
Gaode closed his eyes and reviewed his recent failed construction. He quickly pinpointed the problem: while focusing on controlling the movement of the third Star, the position of the second Star had shifted slightly.
A single misstep would ruin the entire endeavor.
Since the second Star Orbit—which connected the second and third Stars—had already extended, even the slightest deviation in the second Star's position naturally caused the entire Magic Model to collapse.
This was another difficulty in constructing a Magic Model: there was zero margin for error. If a mistake occurred, everything had to be restarted from scratch; one couldn't simply correct the specific error and move on.
"This margin of error is too low," Gaode murmured. Subconsciously, he wondered, "Can the construction process of the Magic Model be optimized?"
If his current thoughts were known to other Mages, they would undoubtedly mock his arrogance and ignorance. Leaving aside the fact that this method of constructing Magic Models had been passed down for countless years and likely had no room for optimization, even if it did, how could a mere Mage Apprentice possibly figure it out?
Gaode, however, didn't share these miscellaneous concerns.
In the world of mathematics, if one method didn't work or was too difficult, changing approaches was a very common practice. Could he determine the positions of all the Stars first, and then connect the Star Orbits?
This thought suddenly popped into Gaode's mind.
As soon as the idea surfaced, he was struck by an epiphany. The more he thought about it, the more feasible it seemed. He even began to feel that this was the correct way to construct a Magic Model.
This way, even if a Star deviated from its original position during construction, it wouldn't lead to the collapse of the entire Magic Model, forcing him to start over from scratch. He would simply need to adjust that specific Star's position in real time.
Compared to the traditional method of constructing Magic Models, this wouldn't just improve efficiency by a little bit. It was the difference between an abacus and a computer.
Gaode had always been a man of action; when he had an idea, he executed it.
The first problem to solve was how to determine the position of each Star.
The Magic Model construction processes recorded in all Magic Formulas involved connecting the Star Orbits while simultaneously determining the position of each Star through relative displacement. They didn't explain how to pinpoint the Stars' positions without connecting the Star Orbits first.
But to Gaode, this wasn't a problem at all. The existing information was more than enough to work with—it was just simple analytic geometry. If he established a Cartesian coordinate system and derived the vector coordinates of each Star, wouldn't he be able to determine their exact positions?
First, he needed an origin point. The origin was the starting point of all vectors. Only by establishing the origin could he determine length and distance, which would in turn allow him to determine the vector coordinates of each node.
In the Magic Star Sea, nothing existed besides the Stars and the Magic Models. However, the Stars were constantly moving; clearly, they were not fixed reference points and couldn't serve as the origin. Although the Magic Models didn't move, they were composed of multiple Stars. How could they serve as a reference point?
If he used one of the Stars in the Magic Model as the origin, it might cause two Magic Model nodes to overlap or create interference from crossing Star Orbits. However, this was easy to solve. He just needed to treat the original position of the first Star as the origin.
Using the origin as the center, he established a classic XYZ coordinate system. Then, he used an ordered triplet array to determine the position of each node in the Magic Model.
The triplet array consisted of three numbers, which directed the path from the origin (the starting point of the vector) to its tip (the end point of the vector). The first number represented how far to move along the x-axis, with positive numbers indicating rightward movement and negative numbers indicating leftward movement. The second number represented how far to move parallel to the y-axis after that. The third number represented how far to move along the z-axis.
Similarly, by following the Star movements recorded in the Magic Formula, he could reverse-engineer the coordinates of each Star.
Gaode stood up, retrieved a charcoal pencil from a nearby shelf, and began writing directly in the blank spaces of the Magic Formula.
The first Star was the origin, recorded as the coordinates (0, 0, 0).
"Forward one, right one and one-third, up one-fourth..."
Left and right served as the x-axis, forward and backward as the y-axis, and up and down as the z-axis.
The coordinates for the second Star were recorded as (4/3, 1, 1/4).
"Forward one-half, right two-thirds, down one-half..."
The third Star moved with the second Star as its starting point. It couldn't be recorded by directly comparing it to the origin, but that wasn't a major issue—it was just basic vector addition.
Through calculation, the coordinates of the third Star were determined to be (2, 3/2, -1/4).
He continued calculating in this sequence.
Before long, Gaode had dismantled the Magic Model for Acid Splash into an XYZ coordinate system and nine vector coordinates, including the origin.
With burning eyes, Gaode stared at the nine triplet arrays on the paper and began committing them to memory. Obviously, memorizing nine triplet arrays was much simpler than the convoluted descriptions in the Magic Formula, especially since Gaode had a naturally high sensitivity to numbers.
In just a few minutes, he had the nine coordinates firmly memorized.
"Let's give it a try."
Since the preparatory work was done, Gaode wasted no time and immediately began his attempt.