The Banach-Tarski Theorem describes how to split a sphere and re-arrange the pieces to make two spheres of volume equal to the original sphere. Turns out it works on dragons, too.
The Banach-Tarski Theorem describes how to split a sphere and re-arrange the pieces to make two spheres of volume equal to the original sphere.
“Alright, Spike, let's double-check the checklist.” Twilight nodded for emphasis.
Spike sighed. At least it was a short checklist this time.
“Apple.” he said.
“Check.” said Twilight. One of Sweet Apple Acres' finest sat on the table.
“Paper.”
“Check.” Several sheets, suitable for writing down any results and observations from the upcoming spell.
“Quill.”
“Check.” For writing down the results with.
“Backup quill.”
“Check.” In case the first one broke.
“Backup backup quill.”
“Check.” In case of quill emergencies. In case of more severe emergencies, well, Twilight had wings of her own now; both Fluttershy and Rainbow Dash (but more emphatically Rainbow Dash) had impressed on Twilight that her wings were not to be thought of as an infinite source of quills, but Spike didn't think they were quite persuading her.
“Ink.”
“Check.” The third thing necessary to write down results.
“Backup ink.”
“Check.” Given the events of the last few years, it was no surprise that Twilight preferred to have lots and lots of fallback plans.
“Backup backup ink.”
“Check.” Though Spike personally felt that she went a little too far sometimes.
“Distraction-free environment.”
“Check.” The library's front door was closed and locked. As was the back door. And all the windows had been firmly sealed by Twilight's magic. There was a large “CLOSED” sign on the front door, with small print underneath that read “Beware: Magical Experiment In Progress”. And after last week's fiasco, Spike had made a point of ensuring that this week's experiment was taking place during school hours, when the Cutie Mark Crusaders were kept in a schoolroom on the far end of Ponyville.
“Annnnnd... that's it, end of the list.” Spike rolled up the scroll. “What exactly is this spell supposed to do, anyway?”
Twilight gave her Number One Assistant a sideways glance. “Weren't you listening when I explained it earlier?” she asks.
“I was listening.” said Spike. “I just didn't understand all that stuff. Something about cutting up a sphere and putting it back together or something?”
Twilight sighed. “It's the Banach-Tarski Reassembly.” she said. “Imagine... um... if you take a block, a perfect cube. If you cut it up the right way, and put the pieces together, you can make, say, a pyramid. Or... eight smaller cubes.”
“Yeeeeesssssss....” Spike nodded, wondering where the bit that he'd stop understanding would come in.
“Well,” continued Twilight, “the Banach-Tarski Reassembly is a way of cutting a sphere into no more than five pieces and putting them together so as to form two spheres, each with the same volume as the original sphere.”
There it was. “But how does that make sense?” asked Spike.
“That's the part that I explained earlier.” said Twilight. “Now, what this spell will do is apply Banach-Tarski to the apple. If all goes as planned, we will have two apples.”
“Ohhhhhhh.” said Spike. “So it's some kind of duplication spell. Gotcha.”
Twilight sighed again. “It's not just a duplication spell, Spike. It's proof of one of the least-well-explored areas of modern topology!”
Spike looked up, and blinked at Twilight.
“...never mind, Spike. Just... don't distract me while I cast it, okay? The spell's a bit tricky.”
Spike nodded. Twilight looked at the apple, and began charging up her horn. She narrowed her eyes, concentrating exclusively on the apple, visualising the equations that would define the points in each of the five pieces which would -
“HIIIIIIIIIII TWILIGHT!” A pink pony popped out of the fireplace, covered in soot.
“Eep!” Twilight jumped into the air, trying to turn around to see what had happened; and midway through turning around, she lost her hold on the spell. “Pinkie! What are you doing in my fireplace?”
“Well, all your doors were locked, silly. And your windows. So how else could I get in here except down the chimney?”
“I was busy with a delicate magical experiment!”
“Is that why you cast that spell on Spike?”
“Spike?” Twilight spun around again, to see her baby dragon floating in mid-air, surrounded by tendrils of magical energy. As she watched, the spell encased Spike in a glowing ball of force, and then... seperated. Five spherical clusters of glowing purple points split apart, and reassembled themselves into two complete spheres before Twilight's eyes. Then the spheres faded, dropping their contents on the floor; a pair of baby dragons, each holding an identical checklist.
“Oh no oh no oh no! Spike! Are you okay?” Twilight rushed forward.
The two Spikes slumped, managing to catch themselves on each other.
“Little...” began the one on the left.
“...tired.” finished the right-hand Spike.
Then both baby dragons fell over forward, asleep before Twilight's aura could catch them.
Spike woke up first, stretching and yawning. He looked over at Spike, still asleep in his basket, and noticed in passing that Twilight had managed to duplicate the basket as well – probably with the same spell. His claws made a slight noise as he put one foot on the wooden floor, and suddenly Twilight was there in a pop and a flash of purple.
“Spike! How are you feeling? Are you alright?” Twilight glanced at the awake Spike, and the asleep Spike. “That's odd, I'd expected that you'd both wake up nearly simultaneously.”
“I'm feeling just fine.” said Spike, grinning widely. “Do you think... Rarity might need some help this afternoon?”
“Er, it's possible.” said Twilight. “But we've also still got to re-shelve -”
“Other Spike can do that.” Spike waved a claw at his doppelganger.
“Well, yes, but wouldn't he want the same thing? I mean, the Banach-Tarski -”
“Shut up, Twilight, nobody's interested.”
Twilight sat down, and stared at Spike in shock. Spike, in turn, tried to stare at his own mouth. It wasn't the first time that he'd thought something along those lines, but it was very much the first time he'd ever said it aloud
Spike was the first to break the ensuing uncomfortable silence. “So...” He began to edge towards the door. “I'm, uh... I'm gonna see if maybe Rarity needs help, okay?”
His claws could be heard clattering down the stairs for a moment, and then silence reclaimed the room. Twilight continued to sit there for almost half a minute, before saying “Not... not interested?”
The other Spike blinked and yawned. “Good morning, Twilight.” he said. “Who's not interested in what?”
“Apparently... you.” said Twilight. “In Banach-Tarski.”
“Well, I wouldn't say I'm disinterested, as such.” said the second Spike. “I just don't understand it.” He yawned again. “This splitting thing – is it going to be dangerous?”
“I don't think so.” said Twilight. “Not for a dragon, at least. I don't know if a pony would survive the process, but if both of you are still alive and healthy, then I don't see any reason why you shouldn't stay that way.” Twilight took a deep breath. “I can undo the Banach-Tarski spell.” she said. “It's pretty straightforward, as long as I do it within the next few weeks. It's really up to you – whether you want me to undo the spell or not.”
“Hmmm.” Spike considered the problem for a moment. “What do you think?”
“I think,” said Twilight, “that you should think on it for a day or two before you decide. Maybe talk it over with the other Spike, as long as he stays in one place for long enough for me to explain that he has that choice. In the meantime, while you think about it, would you like to help me with some re-shelving?”
“Re-shelving.” Spike yawned again, then climbed out of his basket. “One Number One Assistant, reporting for duty!”
* * *
“Hey, Rarity. You're looking beautiful today. Just like every day.”
Rarity turned around, with a smile. “Why, thank you, Spike. And what brings you to the boutique today?”
“Maybe I just wanted to tell you how beautiful you are?”
Rarity giggled. “That's very sweet, Spike. But don't you have duties at the library?”
“Yeah, but Twilight accidentally cast a spell that duplicated me, so I'm leaving all that for the other Spike to deal with.”
“Oh, I see.” said Rarity. “And then tomorrow you'll work in the library while the other Spike gets the day off?”
“Yeeeaaaaaaahhhhhhh... we'll see about that. Anyhow, in the meantime, I was thinking – how about I take you out to lunch? I know this place that does great daisy sandwiches.”
“Why, Spike!” said Rarity, fluttering her eyelashes. “I'd be honoured to -”
“And then,” continued Spike, grinning, “we could come back here and...”
He whispered something in Rarity's ear. Her eyes opened wide, her pupils shrunk to tiny dots, and her face turned bright red.
“Spike! That is not an appropriate request to make of a lady!”
“So is that a 'yes'?” asked Spike, waggling his eyebrows.
* * *
The door to Carousel Boutique opened, and Spike exited, upside-down, with the glow of Rarity's magic around his tail. He was carried out to the middle of the street, and then simply dropped to the ground.
Then the boutique's door slammed firmly shut.
“Well,” said Spike, “that could've gone better.”
With a screeeeeech, a scooter (attached to a small cart) slid to a sudden stop right outside Carousel Boutique.
“Hey, Spike!” said Scootaloo.
“Wanna join us for ice cream?” asked Sweetie Belle, from the cart.
“Sounds good.” Spike jumped into the cart, between Applebloom and Sweetie Belle. “Let's go!”
“You weren't going to go see Rarity?” asked Applebloom, as Scootaloo put her wings into gear.
“Spike? Are you okay?” Twilight's concerned face looked down.
“Yeah, m'fine.” mumbled the baby dragon.
“You've been shelving fiction in the non-fiction shelves for the past ten minutes.” pointed out Twilight.
Spike blinked at the shelf in front of him, then looked up. “Er... right.”
“And in those ten minutes,” continued Twilight, “you've actually shelved a total of two books.” She looked again, and then added “In the wrong order.”
“...huh. So I have.”
“You're clearly not fine, Spike. And it's almost certainly due to the Banach-Tarski spell. Does it hurt anywhere? Are you feeling at all hungry, or thirsty?”
Spike yawned. “Just... a little tired, I guess?” he said.
“Tired? So you need to take a nap?”
“No, not... not that sort of tired. Just kind of... I don't really feel much like doing anything.” Spike shrugged. “So I might as well help with the reshelving.”
Twilight gave Spike a calculating look. Then she said “How about this – once you finish that shelf, we'll go out to Sugarcube Corner to get some icecream, okay? I hear they've got a new line in ruby sprinkles...”
“Meh.” said Spike, waving a claw. “You go. I'll stay here and reshelve.”
Twilight's eyes widened. Spike, refusing rubies? This was clearly a far more serious problem than it appeared.
* * *
“One strawberry with rainbow sprinkles... one lemon icecream with a slice of apple... one chocolate vanilla with ruby sprinkles... and one vanilla with chocolate sauce.” Pinkie tossed the separate bowls of ice-cream onto the table as she named them. “So, is the other Spike gonna come join you?” she asked. “I'm pretty sure that one of you is new to Ponyville, which means one of you needs a welcome party, only I'm not sure which one. Ooooh! More customers! I'll be back in a minute or two, okay?” And with that, Pinkie dashed off.
The three Cutie Mark Crusaders turned, as one, to look at Spike.
“Other Spike?” asked Applebloom.
“Yeah, a spell of Twilight's went wrong.” said Spike. He picked up his bowl and poured the icecream into his mouth, trying to swallow all three scoops at once.
“Spike!” admonished Sweetie Belle. “It's rude to bolt your food!”
“Ig ig?” said Spike, trying to speak around the mouth full of frozen ice-cream. He puffed out his cheeks a little and let out a tiny burst of dragonfire, melting the ice-cream almost instantly and swallowing most of it, leaving the rest to dribble out of his mouth, over his scales and onto his chair. “Anyway, yeah, spell went wrong, two of me, so I've got the day off. I was gonna spend it with Rarity, but apparently she doesn't like me, so...” He shrugged.
“Here.” said Sweetie, firmly passing Spike a napkin to wipe off the ice-cream with.
“Don't mind if I do.” Spike reached out and grabbed Sweetie's ice-cream bowl. A burst of dragonfire melted the contents,and then Spike simply drank it down. With a flick of his tongue, he licked the ice-cream off his face, and then turned to look at Applebloom's and Scootaloo's treats. Both crusaders wrapped their forelegs protectively about their ice-cream bowls.
Spike leapt up onto the table. “SPIKE... WANT!”
Still holding the napkin out in mid-air, Sweetie blinked. “Buh...” she said.
* * *
“It's been half an hour, Spike.” said Twilight, firmly, as she trotted towards Sugarcube Corner. “And you shelved four books. I'm really worried about your electrolyte balance.”
“Yeah.” murmured Spike, seated on her back.
“One of Pinkie's sapphire cupcakes should replace any missing electrolytes.” said Twilight. A twitch of a wing prevented Spike from falling off. “And then you can have some ice-cream, and if you're not recovered after that, then we are going to the doctor.”
“Sure, fine. Whatever.”
“And we need to find the other Spike, too.” added Twilight. “Whatever's happening, it'll be happening to him as well...”
“I don't think so.” said Spike. He pointed ahead, to where Sugarcube Corner had just cracked like an egg hatching. A giant dragon stood over the building's wreckage, ponies fleeing from him in a panic.
“SPIKE WANT!”
“Ohdear.” said Twilight. “Ohdearohdearohdear.” She took a deep breath, overrode each and every one of the instincts that were telling her this was a bad idea, and marched forward.
“SPIKE!” she called out. “What do you think you're doing?”
The giant Spike looked down disdainfully at the purple alicorn. “GO AWAY!” He took a deep breath, and emphasised his words with a burst of green fire in her direction.
“Eeep!” said Twilight, vanishing with a pop. Spike's dragonfire struck the ground where she had been standing, leaving a circle of lightly burnt grass, at the centre of which sat a baby dragon.
“Not cool.” said the tiny Spike, shaking his head. “Not cool at all.”
“Cool doesn't work.” boomed the giant Spike. “I've been trying 'cool' for far too long.” He bent over to stand on all fours, his huge foreclaws thumping to the ground on either side of his tiny doppelganger. “Do you want to know what 'cool' gets you? NOTHING!”
Big Spike lowered his giant head, to look Little Spike directly in the eye. “'Cool' doesn't get you a date with Rarity. 'Cool' doesn't get you an afternoon spent outside a stuffy library. 'Cool' doesn't get you out of a boring lecture. 'Cool' doesn't get you free ice-cream.”
“Free ice-cream made you grow like that?” asked Little Spike.
“NO!” Big Spike reared up and roared – several ponies, who had stopped to look back, decided that on the whole it would be smarter to simply keep running instead.
“I decided it was time!” boomed Big Spike. “Time to do what I wanted! To take what I needed. To finally do all those little things that you never quite did!”
“Fascinating.” said Twilight, from off to one side. A quill pen danced in front of her, writing down notes. “So that's one Spike with no motivation, and one Spike with no restraint?”
Big Spike roared another fireball at Twilight. There was a pop as she vanished again.
“And?” asked Little Spike, raising an eyebrow.
“And they rejected me!” roared Big Spike. “Twilight replaced me. With you. Rarity threw me into the street! The Crusaders ran away from me! YOU were the one who replaced me! I finally started acting like I really am, and everypony HATES me!”
Big Spike took a deep breath. “Spike want... acceptance.”
Little Spike blinked. “It's there.” he said. “We've got it.”
“...What?”
“Do you know why Twilight took me out here?” asked Little Spike. “She was worried that something was wrong with me. She wanted me to get a good meal, she wanted to find you, she wanted to get us both to the doctor. I don't know why you think she replaced you... but she wanted to help you.”
Big Spike narrowed his eyes, and glared at Little Spike. But even as he glared, he began to shrink slightly, from 'enormous' to merely 'very large'. “There's no proof...”
“It's true.” confirmed Twilight, re-appearing again. “I was worried that you might both be suffering from an electrolyte deficiency.”
“But... what about Rarity?” asked Spike.
“I,” said Rarity, stepping forward, “would have been quite willing to accept a date. It was your other suggestion that went too far.”
Very Large Spike shrank down further, to merely Large Spike. “The Crusaders...” he said, in the faintly desperate tones of an angry dragon who's quickly coming to realise that there's not nearly as much reason to be angry as he'd thought.
“You took our ice-cream!” called out Scootaloo's voice, from back in the crowd where Pinkie was trying to keep the fillies away from the giant dragon. “And it's not like we didn't get you your own ice-cream!”
Spike shrank down to merely teen dragon size. “It doesn't matter.” he rumbled, holding on to the tail end of his anger. “Ponyville doesn't need two dragons.” He lowered his head, to glare at Tiny Spike. “You wouldn't last a minute outside here, without motivation.” he said. “I can go and live elsewhere. Somewhere far away, where there's nopony around for me to hurt with my lack of restraint.”
“You don't need to do that.” said Tiny Spike.
“I know I don't NEED to!” roared Teen Spike. “But you deserve -”
“Twilight can put us together again.” finished Tiny Spike
“...what?”
“That's what I was going to tell you.” said Twilight. “When you left. It's up to you... the two of you... whether you want to be turned back. The spell isn't difficult.”
Teen Spike shrank back down into a baby dragon, and sat down. “Yes, Twilight.” he said. “I want you to turn us back. There's only room for one dragon in Ponyville... and I want that dragon to be both of us.”
“And what do you say,” continued Twilight, “um... other Spike?”
“Yeah.” said the Spike still sitting in his circle of lightly burnt grass, with a slight smile. “Do it.”
Spike woke up. He was lying in his basket, at the foot of Twilight's bed. He stretched, and yawned. “That,” he said, “must have been the weirdest dream I ev-”
There was a sudden *pop* of displaced air as Twilight appeared, accompanied by a cupcake. A clatter of hooves from downstairs suggested that more ponies would be joining them very shortly.
“Spike! Are you alright? How are you feeling? Any nausea? Vomiting? Tiredness? Lack of either motivation or restraint? I bought you a sapphire cupcake, in case you need the electrolytes...”
Spike stared at Twilight, and blinked. “You mean, that whole thing about being split into two... that wasn't a dream?”
“No,” said Twilight, “it wasn't.”
“Huh.” said Spike. He explored his memories for a moment, then added “I can remember both mes.” He reached out absent-mindedly to take the cupcake. Electrolytes or no electrolytes, a sapphire was a sapphire – even if these particular ones were a little out-of-season. “How come did Rarity turn up at Sugarcube Corner just then?” he asked.
“I suspect,” said Rarity from the doorway, “that they probably heard you in Canterlot, darling. Big you, that is.”
“Rarity!” squeaked Spike, turning bright red. “I'm, um, I'm really sorry about, it was really very inappropriate...”
“Quite.” said Rarity, reaching up with one hoof to adjust her hat. “Twilight has already explained the side-effects of her spell. So, we shall speak no more of that little incident.”
“Right.” said Spike, nervously.
Twilight gave Rarity a suspicious look.
“In that case,” continued Rarity, stepping into the room. “I shall be expecting you no later than one tomorrow afternoon.”
“Um, er...” Spike glanced around the room, as if hoping to find a handy cue card. What he found instead was Twilight turning her suspicious gaze on him instead. We will discuss this later, you and I promised her frown.
“You did invite me out for lunch, did you not?” asked Rarity. “I believe daisy sandwiches were mentioned.”
“Hey there, sugarcube.” Applejack poked her head into the room. “Don't you ever make the mistake o' thinkin' you ain't accepted around here, okay? You're accepted, and you're loved, and you can stop by the Acres any time you start to doubt that, right?”
“Um, yes. I, I, I do know that.” said Spike. “It's just, well, sometimes... just sometimes... I don't feel that, you know what I mean?”
“Ah reckon,” said Applejack, stepping into the room, “that if there's one pony in town who knows what you mean, it'd be Zecora. And if you need to talk with somepony about it, Ah reckon Zecora's not a bad choice. But if she's by some chance busy, Ah'll always be willing to lend an ear.”
The next pony to enter the doorway was Rainbow Dash. “Hey.” she said, then pointed a hoof at Applejack. “Everything she just said. But less sappy and more awesome, okay?”
Spike smiled. “Okay.” he said.
“And one other thing.” added Rainbow. “You find anypony who's not all accepting, and you tell me, and I'll sort them out, you got that?”
“Y-yeah.” said Spike. He blinked rapidly. There seemed to be something in his eye.
And then Rainbow Dash stepped aside, making room for Fluttershy.
“Um, you rock. Whoo-hoo.” she squeaked, nervously. She shuffled into the room, and hid behind Rainbow Dash.
There was nopony else waiting outside. Spike blinked – given the five ponies already crowded into Twilight's rather small bedroom, he had been rather expecting a sixth.
Twilight snorted, and raised her voice. “PINKIE!”
There was a shuffling of hooves, and a pink pony stood in the doorway. Her mane was half-deflated, and her bounce seemed to have left her. “Spike?” she said.
Spike blinked, and stared. Was this the same mare who had served one of him icecream just a short while ago? “Pinkie?” he asked.
“Oh, Spike!” Pinkie leapt across the room, and hugged the little dragon. “I never meant to almost get you killed and ruin Sugarcube Corner by interrupting Twilight's research! Can you ever forgive me!”
“Uh... sure.” said Spike. He tried patting Pinkie comfortingly on the head.
Pinkie's mane sproinged back to its usual fuzzy state.. “Okie dokie lokie.” she said, all her cheerfulness returned in an instant. “And if you ever need a feeling-accepted party, you can come and talk to me and I'll have you party-sorted in a party-minute, okay?”
Spike hugged back, with a grin. “Okay.” he said.
“GROUP HUG!” yelled Pinkie. She and Spike were immediately ponypiled by five other ponies.
And, at just that instant, Spike knew that he was exactly where – and who – he wanted to be.
