Throne of Magical Arcana

Winter sunlight pierced through the thick layer of cloud, went through the window glass and fell on the astrology maps hanging on the study’s wall. The light yellow halo made the space warm and cozy.

Neeshka’s student, Samantha, wore the same you-still-owe-me-ten-thousand-arcana-points blank expression. "Sir, are you joining the meeting tomorrow?"

"Of course, why not?!" Neeshka was rather pissed. "I’ll teach Levski and Lucien Evans a good lesson! This rubbish should never be sent to the Board again!"

Samantha nodded slightly, obviously not affected by her teacher’s fury at all.

"I’ll tell the coachman to prepare for tomorrow morning and don’t be late again."

Then she picked up the pile of files and left the office, leaving the furious Neeshka alone staring at the paper and the comment from Lucien.

......

In the villa surrounded by flowers, Milina was standing in front of a dressing mirror, looking at her own face which was written with sheer anger. Beneath her feet, there were a few papers torn into pieces. She lowered her voice and murmured, "Lucien Evans..."

In her eyes, Tower Geometry was the only geometric system in the world and there was no other "new geometry" that was essentially different from it. It was just so obvious that Levski’s paper was full of stupid errors, as it conflicted with the real world. All the good words that Lucien Evans used on the paper were complete nonsense. What Lucien Evans was trying to do was to confuse right and wrong, to disguise a pile of cow dung as whipped cream!

There was no doubt that Lucien was now slapping her face, as well as shaming all the Tower arcanists studying mathematics, as the achievements in the field of math were rarely able to receive an appreciation like this. Even her achievement in math, which had made her win Arcana Scepter, never got such a high comment!

Walking away from the mirror, Milina came to her bookshelf and pulled out the papers that she had published. Combining the viewpoints together, she quickly developed a new paper.

Only when she finished did her put the title on: On Parallel Lines.

As a rigorous mathematician and astrologist, she never fought unprepared!

As for the rest of the authorities who had received and read the paper and its comments, they shared the same attitude with Neeshka and Milina. When Levski’s paper was first submitted, they had all contributed to the crucial criticism against this poor arcanist.

......

In the early morning, the haze that had lasted several days finally disappeared. Now the sky was as clear as washed.

Standing in front of the mirror, Lucien leisurely adjusted his black double-breasted suit, white shirt, and light yellow waistcoat. After carefully checking his appearance was proper and decent, Lucien smiled and said to himself, "You gotta change your style today."

Then he put Element, Electron, and Origin on his right thumb, middle finger, and little finger respectively. The flowing light and colors of purple and blue enhanced each other’s radiance, making the rings rather dreamy and eye-catching.

Following that, Lucien took out his shining six-silver-star arcana badge and his Review Board member badge, on which was a hand holding a quill pen, and set them on his left chest.

Meanwhile, on Lucien’s right chest, there was a prominent pattern of a magnificent throne supported by bones. The pattern was there because Lucien had transformed his Immortal Throne magic robe into this black, double-breasted suit.

Although most members in the board had won the highest honor in the fields that they specialized in once or twice, Lucien’s case was still very rare as he had won the most influential awards in three different fields and so many rings. Apart from Fernando, among those who would attend today’s meeting, the best had only won the top awards twice. So Lucien should be able to bring them some pressure by showcasing his achievement. The only problem was that Lucien had not won Arcana Scepter, which was the most authoritative award in Math.

Checking his appearance in the mirror once again, Lucien picked up his black top hat from the hat rack and put it on as he left his villa.

Past the streets, into the headquarter magic tower of the Congress, into the elevator, and up to the fifteenth floor, Lucien arrived at the meeting room at a leisurely pace.

"Mr. Evans?"

When Lucien was about to push open the meeting room’s door, he heard a hoarse male voice calling his name.

Turning around, Lucien saw a middle-aged man who was standing on the nearby patio, looking quite uneasy and restless. The magic robe he was wearing was rather old and out of date, and smoke was slowly rising from the lighted cigarette between his fingers.

"Mr. Levski?" Lucien figured that the middle-aged man was the main character for today. "Why are you still here?"

Levski combed his messy hair with his hand and put on a bitter smile. "As soon as I get in there, the six board members will for sure start criticizing me right away. I’d better just stay here, and... you know, to calm down a little bit."

Compared to the numb and reserved Levski from several days ago, today’s Levski was definitely more encouraged and confident. Now he could at least express himself efficiently.

Lucien understood his feeling, and he smiled gently. "So, are you ready? Are you ready to face their coldness, attack, and sarcasm, and show them your new geometry system?"

Levski frowned, as if seized by the pain in his memory. He said, with less confidence, "I’m ready... but them... I once introduced my paper in front of all the math arcanists in Tower. That was my first time... No one believed in me. There was no follow-up discussion, and all they gave me were nothing more than indifference, ignorance, criticism, and howls of derision. I’m afraid you have to suffer from this with me later, Mr. Evans."

"I have faith in you, and in your new geometry system. So I am not afraid." Lucien said to Levski sincerely.

Levski cheered up again, as he finally found the one person in this world who understood his perseverance and his work. "Mr. Evans, thank you so much. You’re the first person who’s willing to accept my geometry system, and the high appraisal you gave me, even I myself had never described my paper with those words... They really mean a lot to me. Thank you, thank you..."

Levski had devoted most of his time into studying math and arcana, so he was not good at eloquence and could only show his gratitude towards Lucien by saying "thank you" over and over again.

At this time, Fernando, who was wearing his floor-length red magic robe, arrived and asked the two of them to get in.

"Morning, sir. Thank you for coming." Seeing that Levski had sat down, Lucien turned to greet Fernando.

Fernando wore a solemn expression used when discussing serious matters. "I’ve read his paper. Although his geometry conflicts with the world that can be perceived directly, it is still kind of interesting, kind of entertaining."

The Lord of Storm never gave direct praise to anyone.

In the meeting room, there were six other board members besides Lucien. Compared to other conventions which usually involved fifteen or sixteen people, meetings in the Mathematics field were much smaller.

There were not as many math arcanists as those in other fields. Many board members specialized in more than one field and could thus attend different meetings, for example, Lucien was able to attend meetings in Elements and Thermodynamics. However, most arcanists were not willing to dig into math as it was only a tool, which could not directly lead to a change in the cognitive world or an improvement in magic. Therefore, only nine members of the Board in total were qualified for reviewing papers in math.

"Mr. Evans, I’m Neeshka."

Neeshka, who was wearing the grey pointy hat, greeted Lucien with no smile on his face. However, when seeing the pattern on Lucien’s suit and his three rings, his white eyebrows twitched slightly. Subconsciously, he wobbled the black, mysterious scepter in his hand.

Milina and the rest of the members in the room also stood up and greeted Lucien. Although they were not at all enthusiastic, they still remained polite. After all, Fernando, who was famous for his cranky and merciless temper, was also here today, and he was Lucien’s teacher. No one wanted to be furiously shouted at on matters not related to academics.

Lucien also responded politely. He noticed that five out of the six members belonged to Tower. They were all wearing the unique, pointy grey hat, including the two ladies. Somehow the hat looked a bit funny to Lucien.

In Lucien’s opinion, it was obvious that the hat did not suit ladies at all. He purposefully on the opposite side of the long table, facing against the six members directly.

Fernando hated wasting time with words, so he said straightforwardly, "Because Mr. Neeshka, Ms. Milina, and Mr. Evans hold completely opposite opinions for the paper, New Principles of Geometry with Complete Theory of Parallels, written by Levski, we are having the small meeting today. First of all, Levski will explain his paper to us. You board members can raise your hands and ask questions at any time."

Levski silently picked up his paper and walked to the small magic platform in the front. However, just after a few steps, he stumbled against a chair, made a loud noise and almost fell over.

Neeshka, Milina, and the other board members sneered. Levski’s face flushed instantly.

Forcing himself to calm down, Levski projected his paper onto the wall using the magic circle and started explaining his geometry system.

"Yes, Mr. Neeshka?"

Levski’s explanation was interrupted by Neeshka when he reached the first conclusion drawn by deductive reasoning.

Holding the black Arcana Scepter, Neeshka said coldly with suppressed anger, "You tell me, why is the sum of angles of a triangle smaller than 180°?"

"It’s based on the axioms and postulates I mentioned..." Levski pointed at the notes.

Neeshka snorted. "Alright, then you find one for me."

"..." Levski was speechless. This was sheer logical reasoning, and it was not built on a physical model.

Neeshka ground his teeth and said word by word, as if to express the anger inside him. "If you can’t find it, it means that it contradicts with the reality, then your paper is completely a mistake!"

Then Neeshka sat down, giving no chance for Levski to explain himself.

Levski wilted like a piece of limp lettuce, but he still tried to look up. When he saw Lucien’s smile, Levski felt encouraged again.

Fernando lifted his hand. "Keep going."

Levski took a deep breath and continued. After a while, Milina raised her hand.

"Yes, Madam?" Levski asked. He found that he was trembling a bit. Based on his years of experience, he could guess what question was to come.

Holding the Arcana Scepter inlaid with many star-like gems in her hand, Milina stared at the poor, middle-aged man. She put on a cold smile. "So, Levski, tell me, why the perpendicular line of one line does not always intersect with the oblique line."

"It’s also based on the earlier reasoning..." Levski said in a way rather lacking confidence. Lucien could not help shaking his head. This was the very moment that Levski should show his confidence and firm attitude!

Milina picked up the pile of papers in front of her, her smile disappeared, and said, "Show it to me, or find the model."

"I haven’t..." said Levski honestly, "but if we follow the logic reasoning, there is no problem with it."

"This is your problem, not mine. We live in this real world, not your imagination." Milina criticized without mercy.

Then she started reading her own paper, in which every single argument accused Levski’s absurd belief.

Her arguments were stabbed into Levski’s heart like daggers. Levski’s face turned so pale that it looked like he was going to faint at any time.

"That is all," Milina said indifferently. She turned around without sparing a single further glance for Levski, as if he was a clown who intentionally presented exaggerated eccentric papers to draw attention from others.

"Keep going." Fernando said to Levski.

Lucien cast an encouraging look at Levski, which gave this poor man some power, although his voice still lowered.

The board members raised their hands from time to time. Some deliberately distorted Levski’s idea to prove his fallacy, while some followed Milina and Neeshka’s method and objected him based on real life. Since this question had been bothering them for more than ten years, all of the board members were subconsciously using bitter words to attack Levski and his geometry system. Levski’s face turned paler and paler.

He, however, still managed to finish his part politely.

Returning to his seat, Levski closed his eyes, as if he had already seen the final result. He had to admit that there was no solid model to support his belief. Finally, he opened his eyes and cast a sorry look at Lucien.

"The members have explained their point of view during Levski’s presentation," said Fernando, "so now let’s have Mr. Evans explain to us why he believes this paper is of great value."

Lucien adjusted his collar a bit and walked to the platform in the front at a firm pace, with a pile of files in his hand.

"If I were you, Mr. Evans, I would not keep insisting," said Neeshka suddenly, "I understand how much you appreciate subversive theories, but we still have to respect what is true."

He was being this polite to Lucien only because Lucien was the student of the Lord of Storm.

Lucien smiled and looked at him. "Truth is the only thing that I respect."

Judging his expression and tone, it was obvious that Lucien was referring to Levski’s paper.

Milina chuckled, however, the look on her face was as cold as a winter blow. "Mr. Evans, I’d like to remind you that you might be expelled from the board if you make a mistake too obvious on purpose."

"This is what I’d like to remind you all as well." Lucien’s smile was gentle, yet his tone was firm and sharp.

All the board members present felt like something exploded in their mind.

— Was this young man accusing them?

Fernando blinked his eyes, feeling a bit surprised that his student, who was always elegant and polite, would say something so aggressive. It’s so abnormal, and it looked like that someone’s in trouble...

"Mr. Evans, you better first think about how to bring an imagined geometry into reality." said another member, Mabel, who was a serious, ordinary-looking woman. Wearing the grey, pointy hat, she looked like an old nun.

Pinching his face, the middle-aged man with messy black hair named Salgueiro asked in a low voice, "Mr. Evans, I’d really like to know why you think his paper will bring a revolution to geometry like how calculus changed maths. How can you prove it?"

"You never made any achievements in math. I doubt that you’re qualified for reviewing this paper," Neeshka said rather straightforward.

Pissed off by Lucien’s words, the members had become as aggressive as a rooster.

Lucien raised his hand to quiet them down. His voice was loud and clear. "If any of you doubt my capability in maths, please go and talk to the Board after this meeting."

Lucien paused a bit and then raised his voice even higher. "From now on, all things that are not related to this paper shall disappear. You are all board members, not kids!"

Lucien’s sudden roaring made the meeting room silent.

Lucien looked around and nodded to Levski slightly. Then he said in the same volume, "When I speak, no questions, no interruptions. All the questions should be raised after I finish. But during my speech, I am going to ask you questions. Please answer my questions honestly, for the sake of the board member badge you’re wearing and for the starlit sky above you!"

The board members were silent and were at a sudden loss of any excuses to reject Lucien. As a board member, three-time winner of Holm Crown prize, and winner of Immortal Throne, Lucien was qualified to make some reasonable requests. Also, Lucien’s imposing and aggressive manner made them wanted to avoid getting into trouble with him.

"If no one says no, then I’ll take it as a yes." Lucien looked at his teacher.

Fernando said with a straight face, "then do as Mr. Evans asked."

Lucien turned around and operated on the magic circle, so that now only the most basic axioms and postulates were shown.

"Mr. Neeshka, let me ask you. Are these axioms and postulates wrong?" Lucien asked.

Neeshka answered subconsciously, "this is different from the real world."

"Mr. Neeshka, forget about the sunlight outside, forget about the world outside, forget the contents of that paper. Tell me, honestly, are these wrong?" Lucien further raised his volume and demanded harshly.

Shocked by Lucien’s manner, Neeshka took a closer look and found out these were the five postulates from Tower Geometry, the four axioms, and the hypothesis put forward by Levski. So he nodded. "These are correct, but the last one is ridiculous."

"It is proof by contradiction. Don’t you know proof by contradiction?" Lucien asked, his voice full of reproach.

Neeshka’s white brow twitched a bit. Obviously, he just could not say he had no idea what it was. So he nodded. "Then..."

"So anyone? Anyone here thinks proof by contradiction is problematic? Raise your hand!" As if he had been teaching in a magic school facing rows of pupils, Lucien maintained his imposing manner.

The rest of the members all shook their heads.

Lucien then projected a few more lines of Levski’s paper.

"Ms. Milina, following the logic, do you think this part of the reasoning is problematic, based on the premise?" Lucien cast a rigorous look at her.

Milina sneered. "It’s different. You can’t find..."

"Forget those! I said forget them! Only think about the premise and the deductive reasoning! Think about math!" Lucien interrupted Milina, roaring. "Tell me! Is it logically problematic?!"

Faced against Lucien’s roaring, Milina was unsure how to refute him. She carefully deduced from the given premises and then shook her head. "No... Logically speaking, it is correct. No equivalent proposition is used as a condition here."

"Good." Lucien gestured Milina to sit down, and then showed a few more lines.

"Ms. Mabel, following the logic, do you think this part is problematic, based on the premise?"

Lucien kept asking and roaring, over and over again. He kept pushing the board members to only think of the axioms and postulates and logical reasoning.

Answering "No" to Lucien’s questions again and again, the board members faces gradually paled. While the board members had sweat on their foreheads, Levski felt more and more encouraged. Every time when Lucien roared, he became a bit more confident. Meanwhile, Fernando nodded thoughtfully.

With the last a few lines of the paper projected on the wall, Lucien turned to Neeshka and asked in a low voice, "So, Mr. Neeshka, following the logic, do you think this part is problematic, based on the premise?"

His hands clenching tightly, Neeshka could feel himself sweating. He gulped and did not dare to mention the real world again. "No..."

"Good. No." Staring at the six members, Lucien murmured.

Then, suddenly, he roared at the top of his lungs like a violent storm.

"No problem with every single line! Then tell me why this paper is wrong!

"Tell me!"

Startled, Milina burst out, "It conflicts with the reality and what we know..."

"Throw them out of your brain!" Lucien roared, "Tell me, in the sense of pure math, following the logic, based on the premise, where is it wrong?!

"Tell me!"

Neeshka, Milina, and the members all remained silent, having no idea what to say. If, as Lucien said, only in the sense of pure math, the paper was indeed correct.

Levski held his fists tight, his head slightly raising. Closing his eyes, his face was written with complex looks, a mixture of elation, sorrow, pain, and hope.

Somehow, the members started thinking to themselves:

Lucien’s roaring resembled Fernando a lot. He was indeed the student of the Lord of Storm. Maybe... this was another the Lord of Storm...

That was what all the members were thinking right now.

Fernando shook his head and murmured to himself amusingly, "I don’t remember I’ve got a love child."

Seeing that the members were all shocked, seizing the momentum, Lucien put another paper in the magic circle and projected it on the wall together with that of Levski.

"An Attempt to Explain Non-Tower Geometry..." Milina silently read the title of the paper, and then she went on reading further.

One by one, the pages of Lucien’s paper were cast on the walls surrounding the board members.

Neeshka rubbed his brows and said confusedly, "So this is differential geometry that he’s using..."

Mabel and Salgueiro started reading as well. A while later, their faces lost colors. Beads of cold sweat rolled down from their foreheads. They could barely hold their quill-pen.

"This is...!" Levski sprang up from his seat, as if he saw the Goddess of Magic standing right in front of him. Because what was in front of him was the very geometry model he had been looking for! This is the Hyperbolic Geometry model that went beyond normal imagination and experience. This was the most powerful proof!

He cried out silently. After so many years, he finally saw the sun rising in this world, driving away all the darkness. After so many years, he finally saw hope!

Lucien’s paper was not complex. To be more specific, it was in fact very simple. Using stereographic projection on the unit circle, Lucien proved that Levski Geometry was compatible with Tower Geometry. Thus, if Tower Geometry was tenable, so was Levski Geometry!

Lucien’s concise derivation and wonderful proof were full of the beauty of math. This was the most shocking as well as the most solid proof to persuade the members!

Levski cried out loud in his mind: He was not wrong! He was the one who had been right all the time!

At this time, Lucien started speaking again, but his voice had softened. "As is known to all, we are only able to see light within part of the spectrum. To see more, we have to use tools, but the tools also have their limitations."

Since some prerequisites that the paper was based on were not available beforehand, Lucien had to prove them in his paper, which made the paper to some degree more complicated than Levski’s, but the members could still understand. Hearing Lucien’s words, the board members wondered what Lucien was trying to say.

"... So, when light plays tricks in some scenarios, our eyes can deceive us, thus we have illusions. Some illusion spells were created based on this."

The board members nodded, agreeing with Lucien’s words.

Lucien kept going in the soft voice. "Similarly, our ears can also deceive us. We can’t hear when sounds exceed a certain frequency. And under some circumstances, we hear things that do not exist.

"So, our knowledge and experience are limited to the design of our body and soul. We all know how to transform. When we transform into other creatures, do we still feel the same way in this world as we do right now?"

"No," said Levski firmly. Although many transformation spells came from the advancement of anatomy — before the discovery that bats used echolocation, the bat transformation spells all had similar errors — ancient sorcerers still managed to see the world in the eyes of other creatures with the transformation spells obtained from those creatures’ magic patterns.

Looking at the confused board members in front of him, Lucien smiled. "So are we correct? Or are those animals and creatures correct? Obviously, we are all correct. We just have different perspectives. The truth we see is part of the bigger truth. Therefore, our knowledge and our experience are always limited.

"Our imagination is based on our experience, thus our limited experience can easily put constraints on our imagination. As we explore this world further, we’ll see more and more things that go beyond our perception and understanding."

Fernando nodded seriously. He knew what Lucien was trying to say.

Neeshka, Milina, Levski, and the rest of the members were still feeling a little confused. They stared at Lucien, waiting for further clarification.

Lucien raised his right hand and his look became serious.

"So, your eyes can lie to you; your ears can deceive you; your experience can mislead you; your imagination can restrain you."

Lucien paused a bit. Under the board member’s gaze, Lucien pointed at the paper projected on the walls and said in low voice,

"But maths won’t."

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