*Dear Princess Celestia,*

Today I learned that studying friendship is not all about working out in the field. Ever since I came to Ponyville I've been recording my observations on friendship, but today I looked over what I've supposedly been learning and realized it was a disorganized mess! Sometimes I've even learned essentially the same lesson twice and mailed it to you as if I learned something new. What's worse, I've never come up with a novel experiment into the nature of the magic of friendship; my investigations have been entirely haphazard. I've forgotten what I learned back in magic school; science isn't just about a blind categorization of facts, it's also about learning deeper truths. From now on I will try to systematize what I learn, investigating the original insight that led to this work, the equation F = M. I will attempt to generalize observations into testable hypotheses. I will compare my discoveries to those of other scholars. Lastly, I will seek applications for my theoretical work, and be a better friend, a friend who does not need to relearn the lessons of friendship."I can have Spike send it tomorrow morning when he wakes up," Twilight Sparkle mused to Owloysius before signing it. "Hmm..." Her quill snapped back into the inkwell as her eyes wandered back up through her letter and settled on the equation, F = M. "No, maybe it's not finished yet..."

She turned to her research notes and her dwindling supply of parchment. She had spend the last two hours organizing the lessons she had previously learned about friendship, and attempting to render them in the sort of mathematical equations which so succinctly expressed the magical truths she had studied in school. If friendship really is magic, it should be subject to the same logic. She knew the great ponies who had created the math of magic had not had an easy time, and did not expect anything to come to her instantly; but she was still disappointed with her vague scribblings.

She had expected to have an equation ready for the letter, something summarizing and extending her previous letters. Instead she had an inky mess of parchment.

Under the influence of her pink glow, a blank sheet shuffled its way to the top of the stack, and her quill hovered before it. She stamped a little nervous rhythm with her hooves and closed her eyes, impatient that a greater insight was not coming to her, then bobbed the pen magically down to the paper.

*Friendship = Magic*, she wrote stiffly. Twilight had had a teacher in magic school who told her, it is still accomplishing something in science to write what seems obvious; to make it explicit, so that it can be subjected to careful scrutiny.

She furrowed her brow in said scrutiny. Did friendship truly

*equal* magic? There must be some equivalence or Twilight and her friends would not have stopped Luna's misguided plan. But equivalence is a stronger claim than in actually merited.

So she placed a question mark above the "="; but then thought better of it and crossed it out. There were more specific mathematical symbols available.

*Friendship \alpha Magic*?

*Friendship ~ Magic*?

^{[see note]} Twilight jotted these down as possibilities, then found she have more to write about. More proposed equations, and possible tests. As she thought about what she new about the mathematics of magic, she discarded possibilities like Friendship ~ Magic

^{2}. Exercises from school floated back into her mind and she tried working them out under various assumptions about the nature of friendship.

Slowly, a single crossed out statement morphed into a new direction for her research.

Slowly, Twilight Sparkle drifted to sleep.

***

Twilight opened her eyes suddenly to the morning birdsongs and sunshine entering her library. She smiled as she noted Spike snoring, then stood up happily to look over her work. Stiff muscles from falling asleep at her writing table did nothing to dull her pride at making some headway.

She picked up the sheet of mathematics briefly. Next to it, she noticed the unsigned letter to Celestia. What mathematics she had done last night, though exciting, was nowhere near the point of reporting, so she added her usual signature.

*Your faithful student, Twilight Sparkle.* She set the two documents, the letter and the mathematics, on the writing table side by side. To send the letter she needed to wake Spike. To continue her mathematics she needed to go out and buy more parchment, and perhaps another quill.

She though it over and decided her cheery mood was more suited to a walk in the bright morning than to bugging Spike to get up. She could get just enough to carry by herself and be back before he even got up. So she tossed her mane happily and, humming to herself, went out to get parchment.

Spike awoke slowly to the vague memory of the sound of the door being shut. "A few more minutes, Twilight," he muttered without thinking. As he realized no response came, he opened his eyes and sat up groggily. "Twilight?" he said, then eventually got up and wandered groggily around the house gathering his senses.

As he realized she had gone out without him, Spike looked around and noticed the mess on the writing desk; there were sheets of parchment covered with notes, a few books, and off to the side, a letter to Celestia and a neat page of unintelligible intellectual gibberish. Spike realized he had been needed and was asleep. He quickly bundled up the two sheets and sent them burning their way off to Celestia, then hurried out the door to see if he could find her.

***

Celestia was patiently arguing with a thick-headed minister who seemed to think he was doing her a favor by wasting her time, when the letter appeared and she happily took her leave of him. Magically untying and opening them, she smiled as she read Twilight's words. She loved how Twilight had taken so seriously the task of studying friendship, which Celestia had only offered as an excuse for Twilight to stay with her newfound friends in Ponyville. Turning to the second sheet, however, Celestia was at first a bit puzzled, then gasped she recognized some of the equations. She furrowed her brow and her expression became darker as she read. So much was here that she thought Twilight had never suspected, would never glimpse.

What Celestia had thought was harmless research had suddenly come unpleasantly close to secrets she could allow nopony to uncover.

She turned from the letter and called for her guards to ready the royal carriage. Twilight Sparkle's research must be stopped.

*To be continued... because it's two in the morning and I can't just finish writing it.**Note:*By \alpha I mean a Greek letter alpha drawn to look slightly like an equal sign. There isn't even a Wikipedia article on this relatively rare mathematical symbol, but roughly speaking it means "is proportional to." The symbol "~" is even less specific: it means "is eventually proportional to." Twi is trying to quantify here just how little she knows.

(Link to prove the \alpha symbol exists

http://www.physicsforums.com/archive/index.php/t-225342.html )

EDIT: Naturally school has caught up on me, I suddenly need to know tensor products and category theory for this physics class I'm in (can anybody teach me?

) but I will try to continue this... Wednesday maybe?