//------------------------------//
// XXVIII: Limitations 
// Story: Memoirs of a Magic Earth Pony 
// by The Lunar Samurai 
//------------------------------//

	“I’ve got the mail!” I shouted as I stumbled into the lab. It wasn’t easy to open the door while simultaneously carrying a box of paper, but I managed to get it over to the nearest table and gently set it on the floor.
“Ah, good work, Starswirl,” Evenstar said as he strode over and dumped the contents of the box onto the table. “Alright, let’s see what we got today.”
Amethyst arrived moments later, lifting several identical envelopes from the pile and holding them in a grid pattern before her. “Alright, we’ve received 11 letters from Challenger. About 8 are new challenges, and 3 are challenges that have been solved,” she said as she divided the stack into two. “It looks like everyone is ramping up their game for winter.”
“Hmm…” Evenstar hummed as he looked at the two floating stacks. “Go ahead and open them and then update the repository.”
Amethyst nodded then looked to me. “Well, these letters aren’t going to read themselves.” She placed them on the table next to Evenstar’s and lifted a letter opener from the other side of the room. The letters that floated before her turned end to end and raced past the blade, opening each one perfectly.
“Woah…”
“That one took a lot of practice,” she said as she removed the contents from each envelope and let them rest on the table. “Alright. So we need to go through and figure out what Challenger has been up to,” she started as she picked up the first letter from the stack. “I’ll check these solutions to make sure we’re staying up to date with our research.” She pointed to the 8 envelopes before me. “You need to go through those and add them to the repository.”
This repository was a running list of sorts. It detailed every single question that Challenger had proposed for the past decade. It started out as a simple scroll, but in that time its contents had been bound into a book of sorts. Amethyst brought it over to the table along with two quills.
I shook my attention from her magic for a few moments to focus on the letters from Challenger. As you can imagine, they were highly formal and detailed descriptions of crazy mathematical problems that had been proposed. It took a moment for me to dig through the body of the letter before I could decipher its challenge.
It has been anonymously proposed to me that a general angle cannot be trisected using conventional geometric methods. Anypony to prove or disprove this proposal will find his place in my next publication.
“Hey, look at this!” I said as I held the letter up. “It says angles can’t be…” I paused for a moment. Not only did I not know what ‘trisected’ meant, but I didn’t even know how to pronounce it.
“What’s that, Starswirl?” Evenstar asked with a raised eyebrow. “Did you find something?”
“Uhh…” I stalled as I gave the paper to him. I wanted to be able to communicate with him on his level, but I just couldn’t.
Evenstar took the paper and let a frown stretch across his face. “Hmmm…” he mumbled as he set it on the table and withdrew a blank scroll. “An angle can’t be trisected…” he muttered as he drew two intersecting lines. “I wonder…”
He lifted a small box of items from beneath the table and laid them out on its surface. At the time I had no knowledge of a compass or a protractor, all I recognized was the wooden ruler. They were all worn from years of use, but as Evenstar took them in his hooves, I realized how well they had aged.
He carefully began his geometric work, scratching at least a dozen marks all over the page. His brow furrowed as more lines emerged from his pencil. “No, that won’t work…” he muttered as he drew a large X through the confusing mess of lines and drew another angle, this one much larger than the last. It was fascinating watching him work, seeing his mind completely focused on solving such a standard problem. I wanted to be like that, I wanted to be able to use such intelligence to solve problems that others couldn’t find solutions for.
“There’s something wrong here,” Evenstar muttered as he stared at the page. I didn’t know what angle trisection was, so I was unable to see what he had accomplished, but in hindsight I can say that he seemed to have trisected the angle. That being said, there was something in his voice that seemed rather concerned.
“Is something wrong?” Amethyst asked. By now she and I were both enraptured in Evenstar’s work. It had only been a few minutes since he had seen the challenge. “So you proved it wrong?”
“I don’t think so…” he started as he drew yet another angle and repeated his procedure. “There’s something about this method that doesn’t seem to work.” He tapped his hoof on the paper where the proposed solution lay.
“What exactly are you looking for?” I asked.
Amethyst turned to me, but Evenstar shot a furtive glance in my direction. I remember, in that moment, feeling very small compared to him. He was everything I wanted to be, the grand master of knowledge that I could only dream of having.
“He’s looking for an angle trisection,” Amethyst said as she pulled a whiteboard over to her side. “It’s when you can cut one angle into 3 equal angles just with a ruler and a compass.”
“Okay,” I said as I watched her sketch out the angle. “So, it’s impossible?”
“I’m not sure,” Evenstar said as he furrowed his brow a bit more at the page. “But it seems like angle trisection is impossible. The idea that I had before works until you look closely at the math behind it.”
I turned my attention to the page once more, this time noticing the half dozen rectangles that Evenstar had sketched around it. “There seems to be a bit of math that’s in my way.”
“Oh really?”
“Yes, I’ve worked with geometry for decades now,” Evenstar started as he started working out a few equations on the page. “There are very specific mathematical functions that I can perform with just a compass and a straightedge. If you see here,” he said as he pointed at a group of triangles, “This is close. I thought, for a moment at least, that the solution was as simple as marking smaller lines to define as thirds on one of the lines. However, that does not supply an accurate answer. If you look to the algebra, you’ll see that the angles aren’t actually trisected. They’re just close.”
“So, why can’t you do it?” I asked, trying to decipher the cryptic writing on the page.
“I think the answer lies in the way we bisect angles.” Evenstar took his tools and, in a few swift strokes, drew another angle and then a line directly through its middle. “See this?”
“Yeah.”
“This is a bisected angle. I can do that again and again and again, splitting each arising angle into yet another angle, but there’s a pattern to it.”
“A pattern?”
“I’m taking half of the first angle. Then I take a half of that. Then, after that, I take yet another half. I’m just multiplying the size of the angle by a half each time.” Evenstar looked to me with a slight smile. “I think I know what’s wrong.”
“What might that be?”
“The only thing I can do to angles is cut them in half. There’s no way, just using multiples of two, to get a number divisible by three.”
I’m not sure why that statement struck me so deeply. There was something so nonchalant about it, something that seemed so steadfast, that I felt there was no reasonable way it could be true. “But how?” I asked.
“How? That’s just the way math works. In fact, any number multiplied by itself will never be divisible by the following number.” Evenstar 	looked back to the page with his impromptu proof.
“But…” I started, not sure what I was asking. I wanted to break that rule, to tear through the idea that I could be bound. There was something about the structure of math, the inherent truths and falsities that challenged me on a fundamental level.
“That is simply how it works,” Evenstar started without looking up from his page. “There are fundamental rules about math that just don’t change.”
I didn’t voice my consternation, I wasn’t about to question my mentor, but it didn’t sit well with me. I wanted, to be completely unbridled, limited only by the world I could imagine. It was fundamentally against everything that mathematics was, but the thought of such rigid boundaries was enough to make me despise the idea that I could be limited.