//------------------------------// // Prologue // Story: The Banach-Tarski Dragon // by CCC //------------------------------// “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.