Shears

by Antikythera

Thorns, 6

Manually attenuating the frequency of a thaumic wave can produce diverse effects, via interactions with both the target's magical absorption spectrum and the spectra of air and other surrounding matter that might scatter the beam. For an illustrative example, consider shining real-frequency (i.e. electromagnetic) thauma into a red pane of glass. That the pane appears red to our eyes means that red light is reflected by the pane, while green and blue shine through unimpeded. Of course, actual EM absorption spectra are significantly more complicated. Typically, there are primary reflection regions centered around the colors constituting its apparent color. However, there are also smaller peaks and bands of reflection all over the spectrum, small enough that they contribute little when blended together with the larger regions.

Complex-frequency (i.e. magic) thaumic waves behave by fundamentally the same principles, but in practice, the mathematics behind the phenomenon work out quite differently. We will consider purely imaginary frequencies at first, but note that this is an oversimplification; unless the magic is invisible to the naked eye, the magic frequency has both real and imaginary components.

Now, consider this question: What does it mean to rotate by an imaginary angle? Multiplying a complex number z by e^{i\theta} rotates it by \theta haydians in the complex plane. What if we plug in \theta = i\omega, with \omega real and positive? We're then multiplying z by e^{i(i\omega)} = e^{-\omega}; we're not rotating z at all, but rather scaling it. Rotating through an imaginary angle is scaling. Let's consider a thaumic wave with frequency i\omega, with \omega real and positive. Instead of oscillating in time, it decays in time. This is why magic bursts lose power so quickly when traversing much distance at all. Of course, a purely real-frequency thaumic wave (light) is expected to decay as well, but this is because air (or any medium) scatters and diffuses the beam. Imaginary-frequency thaumic waves are also subject to this decay, meaning they still decay strictly more quickly than a real beam of equivalent amplitude.

We can now see exactly why it's impossible to emit a thaumic wave with a negative imaginary component: Such a wave would grow exponentially in power, attaining arbitrarily high energy if afforded enough distance to traverse before being absorbed. This violates numerous conservation laws, including energy and linear momentum. However, this fact is a solid data point for time being a complex dimension (i.e. differing from spatial dimensions by a factor of i), and for identifying time reversal symmetry with complex conjugation symmetry. The philosophical reader is directed to Time and Space by Dreamcatcher.

The real absorption spectrum of a material can be described as a function from the positive reals to the interval [0, 1], quantizing what proportion of the wave's energy is absorbed (and how much is reflected). Though the complex numbers admit no total ordering, we will call a complex number \theta = \sigma + i\omega positive if both \sigma and \omega are positive, merely as a convenient shorthand. Thus a positive complex number is one that occupies the top-right quadrant in the standard plane depiction. Then the full (2-D) thaumic absorption spectrum of a material can be defined as a function from the set of positive complex numbers to the interval [0, 1]. Call this function \alpha(\theta). To first order, this absorption function (or, really, its complement, the emission function) typically factors into real and imaginary parts, i.e.

\alpha(\theta) = 1 - (1 - \rho(\theta))(1 - \iota(\omega)).

Note that, since almost all materials show little-to-no magical resistance, the complex component of the absorption spectrum of a mundane material (e.g. wood, stone) is almost completely full. In the factorizable case, a measurement of the 2-D spectrum can be made by measuring each of the 1-D component spectra. However, there are numerous higher-order perturbations that can affect \alpha, such as the Haymahn effect and various multipole interactions. These perturbations are more prevalent in some media than others; for a more robust treatment, the reader is directed to Haymahn's The Wave Function.

Twilight makes a mental note of this reference, but knows she's sinking too deep into the theory. She skips forward a few chapters to the section on magic signatures, hoping for an in.

Every unicorn has what we call a frequency signature. This is a spectrum (i.e. a function from the positive complex numbers to [0,1], typically denoted as \Psi) that is characteristic of her magic. The restriction of \Psi to the positive real branch of the complex numbers is what determines the visual color of a unicorn's magic, but that is not the totality of information embedded in a spell; each unicorn also has a distinctive magical frequency signature as well. Like the absorption spectra of materials, a unicorn's thaumic radiation spectrum is typically first-order factorizable into real and complex components, but there are exceptions; famously, Princess Celestia's emission spectrum follows a 2-D radial Trot Song distribution, peaking around yellow in the reals. Star Swirl's spectrum exhibited an extremely irregular pattern that many believe was fractal in nature, qualitatively similar to the so-called Mewlia sets. Finally, the only known non-unicorn whose magic follows qualitatively similar mechanics to ponies', Discord, had a spectrum which defied all attempts at logical description and was only ever characterized experimentally. Of course, he would hardly sit still long enough for such an analysis to be performed with much rigor. Most data about Discord's spectrum came from retrospective analysis of whatever chaotic debris he left lying around instead of direct horn measurements.

Frequency signatures are typically regarded as irrelevant to everyday life. With one exception, there has never been found a correlation between spectral properties and magical parameters describing a unicorn's abilities. Neither raw thaumage capacity \Theta, nor spell power \Pi, nor recharge rate \Xi seem to be correlated with, say, relative concentrations in the spectrum. The exception mentioned above is that the integral of a pony's spectrum over the positive complex domain (typically denoted \Gamma, the thaumic modulus) is positively correlated with \Pi and negatively correlated with \Xi. However, measurements of \Pi and \Xi are significantly more useful and thus more ubiquitous, so \Gamma is typically computed from them in a unicorn diagnostic and not the other way around. Since two ponies with identical \Gamma but wildly different \Psi exhibit no measurable difference in abilities, knowledge of one's signature does not really afford the pony any useful information.

For this reason, administering a thaumic spectroscopy diagnostic is rarely done in practice; at best it is useless, and at worst it is a liability if that pony ever somehow finds herself in certain situations. However, with enough diligence and determination, a unicorn with academic interest or mere curiosity can usually find a thaumic spectrologist willing to perform the analysis for the right fee.

Signature knowledge may not be particularly useful in daily affairs, but there is a context in which it is of paramount value: magical combat. Though Equestria has been at peace for several centuries, in times gone by, the spectra of powerful enemy mages were frequent targets of espionage. The reasoning is this: when one unicorn's magic interacts with another's, the resulting behavior is partially dictated by the spectra of the pair. We will give a simplified example, in which we pretend both ponies' signatures are purely real and that their spectrum functions are Lilac delta functions. Say Allspice and Basil are locked in magical combat, and Allspice erects a projection around her flanks and front. Her magic (and her projected barrier) are pure green. If Basil's signature is also pure green, his vivification bursts and beams will reflect helplessly off of her shield. However, if his signature is pure magenta, the RGB complement of green, a beam of raw magic will pass through the barrier entirely, routing her defenses. If his signature is pure yellow (which has green as a component), some of his attack will penetrate and some will not. Namely, the pure red component of his spell will totally transmit through the shield, and the pure green component will totally reflect. A common first guess is that \frac{1}{2} of the power of the beam penetrates, but this factor is known to be \sqrt{\frac{1}{2}} instead. See appendix B for a derivation of this factor in the general case.

While our toy scenario is instructive, in practice, what happens when two unicorn mages cross horns is nothing like the above. Since spectrum signatures are typically far more "full" than "empty", the transmission factor is rarely more than a few percent, and one unicorn can almost never directly pierce the shield of another to any useful degree. However, this percentage can make a pivotal difference when compounded several times over. This is why unicorn combat brigades are often segregated by magic color, a phenomenon that poet Silent Spring famously dubbed the "blood rainbow."

There is another way advance enemy spectrum knowledge can be leveraged to (much greater) effect. Relying on there happening to be a mage in your forces with the right spectrum to counter an enemy is entirely up to chance, and only marginally useful in any case. But, on the other hoof, enemy spectrum information can play a pivotal role in the efficacy of assassination attempts. There is no known way a unicorn can alter her signature to counter her foe, but proper planning can utilize the enemy signature to extreme advantage. For example, consider suspending a vivification beam in a gemstone. Though the vast majority of mundane materials have almost completely full complex absorption spectra, gemstones are the archetypal exception, each type of gem having its own characteristic empty bands. As such, vivifying a gemstone of the right composition can be used to "filter" the spectrum of the beam, removing component frequencies (and of course weakening the beam). This suspension can then be released into a gem of a different type. With enough resources, this chaining can be used to finely attenuate the frequency of a beam. If this sequence is repeated enough times to bring the stored spell back up to its original power (or often higher), it can serve as a perfect, pointed weapon targeting the enemy's Aquilles' heel. For an undercover agent, these aptly named "killstones" can be easily passed off as any other spell suspension, to be unleashed after infiltrating to proximity of the target. This tactic is effective even on individuals of extremely high value and standing, who often have a dedicated unicorn continuously shielding them in times of war. It can also be instrumental for would-be-assassins who are significantly weaker than their targets. For a sketch of the methodology of a frequency signature diagnostic, refer to Appendix D.

The end of Twilight's quill has somehow meandered into her mouth. But upon reading that final line, and feeling something faint begin to shine, she retrieves it from between her teeth and sets it to parchment immediately, the other half of her magic beginning to turn the pages.

"Spike? Are you ready for a break? I've found a matter that's more pressing." Twilight gets up from her seat and walks over to the baby dragon, who looks up from Kingdom's Pride with a smile.

"I thought you'd never ask, Twi. This is some of the driest stuff you've ever made me read."

Twilight wants to make certain choice remarks, but she reminds herself he waited until he was done to complain. "Well, is it at least useful?"

"Eh... Maybe, sort of? Most of these books don't know much more than we do, but I wrote down a couple things that sounded promising. Oh, and Elements of Mystery was just a book about party planning—"

A shrieking gasp rends the studious air as Pinkie Pie bounds down the stairs. Wait, was she already in the hou— "That's my FAVORITE BOOK!" she howls. "Hullabaloo is like my spirit animal, if I weren't a pony too, or if she were, like, a squirrel or a gecko or—" she gasps again "—an alligator!! And some alligators know how the hay to party, lemme tell ya!"

"Heya, Pinkie!" Spike greets her, unfazed.

"Uh, hi, Pinkie," Twilight greets her in turn. "You're actually exactly who I want to see right now, believe it or not."

"I belieeeve it!!" Pinkie screams, craning her neck at an obscene Pinkie angle, for some weird Pinkie reason.

"...Eheh." Twilight doesn't roll her eyes. "Nice work with the school bell, by the way. Did Time Turner seem to understand his instructions?"

"He's got them down! It's like he was born to do some random task in exactly regular intervals forever! He got his kids to help him, too. He's got two totally adorbs foals, Grains of Time and Spiral Bevel..."

"I'm glad. It's not a one-pony job," Twilight states, trying not to start kicking her hoof. "Anyway... I was hoping you could connect me to somepony else in Ponyville this time."

"Yeah? Who is it? Somepony cool?"

"Uh, probably? I need to get in contact with a carpenter and a lapidary."

"Hmmm... well, Ponyville doesn't have a carpenter by trade. No Square Edge or Cut Corners here. But the Apple family has to rebuild their barn like twice a year, so they've become pretty good at woodwork. Just ask Applejack! I don't know what a lapidary is, though. Is it somepony who drinks a lot of milk?"

"It's somepony who knows her way around a gemstone," says Twilight, keeping her tone level.

"Ohhh!! You can just talk to Rarity! You know, one of her super special talents is finding gemstones! She practically has a hoard of them for her super secret fashion designs!"

"...Huh. That's... somewhat convenient, since I already know both those ponies, but they're both off on tasks. I'll have to catch up with them when they stop back by, but..." She pauses. "I'm not convinced it's smarter to wait."

"Actually, Rarity's at her house right now! I was just checking up on you and Spike, then I was gonna go help her sort through her coats. You wanna come?"

Twilight sits on her haunches and thinks for a second. "...No. Spike does." She retrieves her notes from her table and shoves them into his lap.

"Huh?" he grunts.

"You're going shopping. Specifically, for the gemstones on this list," she says, ripping off the bottom of the top page for herself. She moves to take a coin pouch out of her travel bag lying on the floor, still yet to be fully unpacked. "Here's the cash. Should be more than enough. Actually, let me grab a few hundred bits," she says, and pulls out a few larger coins, shoving them into a second bag. Money isn't going to be worth very much for very long; better to cash in right away than wait until nopony wants it anymore.

"Is this... really the time to buy gems?"

"It is. I'll explain when I have a second. It's convenient that we know where Rarity is, but that doesn't justify putting off talking to Applejack. We've got work that needs to be done immediately; it may already be too late. I'm going to go find her." She knows Spike won't complain about an excuse to see Rarity, anyway. She trots toward the door.

She almost opens it when she reaches it, but something catches: she can't stand it when she's reading a story and a character runs off for some mysterious purpose with hardly a word. Talk to each other, buffoons, she silently begs the page. Communicate. It's an obvious trick to conjure intrigue and get the reader to flip just a few more pages. But what place does intrigue have in real life? Everypony here is going to keep flipping, no matter what. Has she read so many stories that stiff, tired drama runs through her veins, even at a time like this? Is that why she finds herself drawn to a classic protagonist's pitfall? She's spent so many daydreams relishing in how much better she knew she could do in their place. No. Real life doesn't have a protagonist. I'm smarter than this.

"...I want to get out the door as quickly as possible, but I'll say a little bit about why," she continues. "I think one of our smartest moves right now is to get a hold of Princess Luna's magical signature. I need some equipment to perform a measurement of it, so we can possibly use it against her."

Pinkie and Spike exchange a brief glance. "Okay, Twilight! Sounds smart, as usual! We'll let Rarity know it's for a good cause!" says Pinkie Pie.

"...Okay, Twi. Sounds like a plan," Spike agrees. "You wanna just meet back here?"

"Do you have any books left to read?"

"Two more after I finish this one," he says, now standing.

"Then sure. It'll probably take me longer to finish talking to Applejack, so just finish up reading the last few when you get back. If I'm not back by the time you're done... I dunno. Do something productive."

"Like take a nap? It's been ages since I got a full night's sleep."

"Um... yeah, if there's downtime, that's not a bad use of it, honestly. I..."

Pinkie and Spike look at her, expectantly.

"I'm pretty tired, too. I might sleep after I finish the diagnostic."

Pinkie Pie lowers her ears. "Diagnostic? Is somepony sick?"

"...Nevermind. For now, it's time to go. See you later, girls," she says. She lifts one of the two candles from its rest affixed beside the door, and steps out into the night.

Twilight has entered a very different Ponyville than the one she flew into yesterday morning. Even after she blasts alight the modest candle, she can only see a few hooves in front of her face. The lonely moon is nowhere to be seen behind the growing cloud cover, which mangles her vision so thoroughly she can barely read the Golden Oaks Library sign hanging directly over her head. Every now and then, she catches the darting shadow of a weatherpony overhead, sometimes towing a cloud into town from the surrounding hills, sometimes on her way back out for another.

Looking straight ahead from where she stands, Twilight can see eight or nine dull red pricks of candlelight bracing the outline of Ponyville Main Street, but none except her own is close enough to add any definition to the darkness. They sketch a coarse framework of where buildings are and aren't, and offer nothing else. Checking her flanks, she sees quite a few more, but the infrastructure isn't so regular in other directions, and she's not familiar enough with the town to know whether she's looking down another main road, or catching a glimpse of a side street running perpendicular, or locking eyes dead-on with a lit window. This tableau of bleak glow and drab facade is all there is to perceive: the sky is black and featureless, there's not a sound to be heard except a low rumbling wind, and there's certainly nopony else out sharing the view.

But by far the most immediate and sinister facet of the dark jewel Ponyville has become is the cold. It's easily below freezing, which Twilight did not expect this early. How long did we spend reading? I didn't notice the schoolbell ring again... but I could have missed it. She fantasizes about going back inside to retrieve her winter coat, if only it were there waiting for her and not crammed in a Canterlot closet. Just idly standing in the doorway, she begins to feel the numbing pain.

Time to go. There's no way Applejack isn't done talking to the mayor yet, so if I didn't hear from her, she must have gotten the go-ahead. If I were her, I probably would have prioritized salvaging food from city hall, so I guess I'll go see if she's done.

Despite herself, she makes a vague gesture to reposition the coat she isn't wearing, bringing it closer toward her core. It's going to be a long night, she thinks, not for the first time, and not for the last.