“So… what now?” Amethyst asked as she stepped to my side. We had deliberated for hours over changing the MMU only to find ourselves more confused and frustrated than before.
“Well… Let’s start from the beginning…” I said for what felt like the hundredth time.
“Starswirl, if you’re going to try to change the dimensions of the MMU it just won’t work. We’ve tried this at least a dozen different ways but every time we end up hitting the same problem.”
“But why would there be a random arbitrary number associated to a side length? It just doesn’t make any sense.”
“Maybe Evenstar knows something we don’t,” Amethyst offered as she glanced to the door.
I gritted my teeth. “He said he wanted to be alone, but we’re so close…”
Amethyst stepped closer and laid a hoof on my side. The gesture melted my building rage long enough for me to consider her plea. “Why not work on something else?”
“What else is there for us to work on?” I asked as I turned to her. “The MMU is step one…”
“It’s a step…” her last word lingered as she looked to the sky beyond the window. “But it’s not the only step. We’re not running a race, Starswirl, we’re figuring out a puzzle.”
“If you’ve got an idea then say it,” I frowned.
“Alright, alright,” she surrendered as she walked to the table with her research. “The MMU is only part of the equation. The other half is mathematics: the dome itself.”
As she lifted the gridded paper from the table, I finally realized what she had been hinting at. “You’re right…” I muttered as I stepped to the page. “We still have to figure out the shape.” I leaned to my right to bring her face back in view. “But we won’t be able to do much, will we? I mean the MMU is what everything else relies on, what can you do without it? Then again I'm hitting a wall here, so I guess it can't hurt to try.”
We stepped to the table as Amethyst laid down the sheet. I watched for a moment as she studied the grid between us. She was so intent, so determined to see our crazy idea through that I couldn’t help but feel a connection to her. We wanted the same thing, we were making the proper steps, and we were doing it together. There was comfort in that thought, that concept of being with somepony else.
“Hmmm,” She hummed as she drew closer to the page. “I’ve always tried to use the MMU… I don’t know what to do now that it’s gone.”
“Well…
“I’ve always tried defining the shell based on the grid, but what about…” her words trailed off as she scrutinized the page even closer than before.
“What about what?” I asked as the quill beside me rose from its well.
Her nose nearly touching the page, she drew a perfect cross through the center of the blocky circle, dividing the image into quarters. “It’s an axis…” She lifted a pencil from the table and began tracing through the centers of the boxes. “And this circle, at least it’s top half… It’s continuous… It’s a function.”
My eyes darted to the page as I watched her revelation unfold on the page before us. I couldn’t quite understand why I was so captivated by such a seemingly obvious fact, but I stood enraptured as my mind overthought its implications. “What’s the equation?” I asked as she scrutinized the line.
“Well, I wanted it to be a sphere, so this should be the square root of the x value’s square and the y value’s square.”
It took us a few minutes to check to see if her math was correct, but once we did, another revelation began to dawn on me. “If we draw another line next to it, we’ll have an area… do you think we could use that for volume?”
“I think so,” she said as she drew a smaller circle on the page. “But if I’m going to cast this I have to know the exact volume. That’s why I always used cubes.”
“An exact volume?”
“Yeah. That was one of the limitations of the MMU, you can’t cast spells without knowing how much magic you can use. It’s not like filling up a cup with water and the extra spilling over the side, it needs to be perfect every single time. Counting the cubes worked just fine, and that’s why Evenstar made it cubic to begin with. Still…” Amethyst’s voice faded off as her gaze drifted to the window. “We have that equation though. If I put any number in it, I’ll get an answer every time.”
We grew silent as we both pondered the circle before us. We knew what we needed: an exact solution to an approximate problem. I don’t know how many scholars had pondered the same questions we were asking, but the barrier of perfection lay in between us and our solution as it had for millennia.
“But, if we get a really close estimate with cubes, would we get something that you could cast?”
“Nope, it has to be exact.”
“So then we can’t do it,” I started as I looked at the circle. “Pi is just an approximation, unless we know it exactly there’s no sense in using a circle… right?”
Amethyst looked to me once more as a frown spread across her face. “So then do we just give up?”
“No, but the circle itself, we need a replacement. Something that’s like a circle but doesn’t involve pi.”
“A parabola,” Amethyst whispered as she stepped to the bookshelves. “It’s a conic section. I think I can make the math work for it.” She withdrew a small book from the shelf and flipped through its browning pages. A moment later she furrowed her brow and she brought the book before my muzzle. “See that? It’s just y equals x squared.”
“So there’s no pi?” I asked, the spark of hope glowing brighter once more.
She shook her head. “If we set y equal to the negative of x and then add, umm… two or something, then we’ll have a half dome!”
“Perfect!” I exclaimed as she sketched a grid on the chalkboard before us.
“So we’ve got this dome and if we add another line to it we’ll get our area where my magic will be cast.”
“And then all we have to do is-” I caught myself before I could finish my sentence. The next step was to figure out how to accurately calculate that volume, a task that was much easier said than done. Amethyst’s excited expectation slowly melted to mild dejection as she realized I had no way to finish my sentence.
“So now what?”
I inadvertently looked to the chalkboards to see if there was a solution hidden right before me. Between the cracked board, the first half of the analog equations, and the thousands of proofs that walled the rooms, offered their aid in our endeavor. “I don’t know…” I whispered as my gaze drifted to the floor. “Is there any way to estimate the volume? Maybe we could get some clues from there.”
She rubbed her chin as she thought for a moment. “There is a way to approximate the volume underneath a curve, but it’s very messy.”
I shrugged. “That’s about as good as anything else we can do right now. How does it work?”
She stepped to the board once more and drew a rather strange E shaped symbol on the board. “A researcher a few years ago made a notation for adding up a bunch of numbers based on one equation. He called it summation, but it’s not very commonly used. The only reason I ever had to use it was because I was looking for ways of making the math of my shell spell a bit easier to do.”
She drew a number above and below the symbol. “So the bottom number is where you start your calculations, and the top number is the number you end at. Then, all you have to do is systematically work the function and add the results.”
I was enraptured as I watched her begin to add the functions to one another. It was a remarkably simple process but the result was an incredible revelation to me. I hadn’t ever thought of using mathematics as something to systematically perform mathematics before, but once I saw it used in such a way, I couldn’t help but sit aghast at the concept. I had an idea, one that involved some strange notions, but it was only just budding as she drew that equation on the board
“So,” she said as she faced me. “Does that make any sense?”
“I- I think so,” I said as I cocked my head to the side. “So we just do that function a hundred times until we finally get a massive number at the end?”
“Yep, and I can do this a lot faster for magic because of the notation.” Her eyebrows rose as a hopeful spark rose in her eyes. “So do you think we could use it?”
“I… well maybe,” I started as I looked to the parabola on the board.
A smile crept across her face as she stealthily placed a period before the bottom number. “If we do this, the calculation takes longer, but the end result has a higher resolution.”
That’s when the realization hit me: the bottom number controlled the resolution along with the starting point. If we could choose the resolution, then maybe we could incorporate a limit into the equation. Obviously I hadn’t articulated my thoughts to the scale of that revelation quite yet, but the idea itself was forming in a much more abstract way in my mind.
“Mhm,” she hummed with a smile and a nod. “I’ve seen some other researchers use this to find the approximate areas beneath curves. It’s not exact, but well…” She looked at me and shrugged to finish her sentence.
“Well… what?” I asked, trying to coax the sentence from her.
A wry smile accompanied her gentle gaze. “You’re the anomaly, Starswirl.”