• Published 11th Jul 2014
  • 1,574 Views, 22 Comments

The Banach-Tarski Dragon - CCC



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.

  • ...
2
 22
 1,574

Epilogue

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.”

“Oh! Oh. Yes. Right. Lunch. Tomorrow. Yes. Right.”

“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.

Comments ( 18 )

Great little story. Everything went better than could be expected, as scary as the spell's mechanism sounds. Cut into pieces? Ow. :moustache:

Short, but to the point. Nice job.

Maths! Yay! Though I'm relatively certain that it doesn't cut the volume up, just the surface. So if you flay Spike, you can make two dolls of him, but you can't make two Spikes.

Oh, and just saying, the theorem doesn't apply to Spike, since he's a torus, not a sphere. Digestive tract and all that.

CCC

4679018

That's why it's (probably) lethal to ponies. But dragons are remarkably hard to kill, fortunately for Spike...

4679020

:yay:


4679053

Though I'm relatively certain that it doesn't cut the volume up, just the surface.

No, I'm pretty sure it's the volume. Though I admit I have only the vaguest understanding of Banach-Tarski (and most of that comes from a perusal of the Wikipedia page on the subject) - but the Wikipedia page does specify that certain points within the sphere (including the central point) are included.

Oh, and just saying, the theorem doesn't apply to Spike, since he's a torus, not a sphere. Digestive tract and all that.

Ah, now that's a much more valid objection. To which I can only respond that the sphere that was duplicated included Spike and his checklist and the quill he was holding and the contents of his digestive tract and quite a lot of air. So while Spike is topologically a torus, he can still be encased within a sphere and then that sphere can be duplicated.

(I'm not sure if Banach-Tarski works on toruses or not. The Wikipedia article claims it works on "all 'reasonable' solid objects", but then gives two examples that are isomorphic to a sphere, so I don't know if 'reasonable' is intended to include toruses (torii? toruseseseseses...es?) or not)

4679218 Topology isn't my strong suit. I'll trust Wikipedia here and be like, "K brah, totes does the volume".

Ooh, I misunderstood the way it worked. That's totally cool then. And I'm a derp.

So if the resulting dragons are the same size as the original, does this mean that the duplicated Spikes have half his original density? Thus they're half-dragon, half-air?

Speaking of the personality split, were those qualities chosen along a right-brain, left-brain (emotional, logical) separation?


In Prologue:
As she watches, the spell encased Spike—

watched

In Confrontation:
(1) The rest dribbled out over his scales.

You don't have to break the flow of the story to footnote this. It works just fine immediately after the sentence that references it.

CCC

4679738

So if the resulting dragons are the same size as the original, does this mean that the duplicated Spikes have half his original density?

It's worth bearing in mind that a dragon's body is capable of producing mass from nowhere. When Spike experiences a greed-growth burst, he ends up not merely bigger, but heavier. It seems reasonable to consider that Spike's body can use a similar mechanism to recover lost mass in a case like that.

Speaking of the personality split, were those qualities chosen along a right-brain, left-brain (emotional, logical) separation?

No, but that would have been a good idea. The original prompt was based on an episode of Star Trek, where Captain Kirk was split into two Kirks by a transporter accident (one good and one evil).

Spike's personality split here was along the lines of one Spike having a lot of motivation and no restraint - so that he does and says all those little things that one thinks of but doesn't actually do - and one Spike with a lot of restraint and no motivation, so that he has trouble doing anything at all and is nearly incapable of putting his wants before anyone else's.

watched

Whoops, fixed.

You don't have to break the flow of the story to footnote this. It works just fine immediately after the sentence that references it.

I almost left that line out entirely... hmmm. I'll think about it.

Best "transporter accident" pony story I've ever read.

What? Spike was separated into sets of zero-dimensional points, then reassembled. Leptons are indivisible point particles. The analogy is sound.

Also, looking at the Wikipedia article, I'm a bit disappointed you didn't do anything with the "chop up a pea and reassemble it into the Sun" explanation. I'm sure Celestia would be less than thrilled with that one. :raritywink:

In any case, thank you for a delightful pony-flavored exploration of exotic topology.

Cute little story, if shorter than expected. I think this does the show thing where they take an idea big enough to hang a great deal of content off of, and use that to explore a single facet of it.

I'm vaguely disappointed that in the epilogue Twilight doesn't clone herself, but that might have been more silly than you were going for. Anyway, nice job.

CCC

4689177

Best "transporter accident" pony story I've ever read.

The original prompt was actually based on an episode of Star Trek, where a transporter accident led to Kirk being split into good and evil halves...

So, well spotted. :twilightsmile:

I'm a bit disappointed you didn't do anything with the "chop up a pea and reassemble it into the Sun" explanation.

Well, that would just lead to a pea the size of the Sun...

Unless it's Discord doing the chopping-up, of course, in which case anything goes.

4689231

Personally, I think that this trope is a better fit... but if the second Spike really needs a moustache, I'm sure Twilight can provide one.

4689258

I'm vaguely disappointed that in the epilogue Twilight doesn't clone herself, but that might have been more silly than you were going for. Anyway, nice job.

Hmmmm. That... wouldn't be out-of-place in a sequel, perhaps. The Banach-Tarski Princess... aaargh. I have too many stories to write.

CCC

6509969

You can never fully Pinkie-proof your house. NEVER!!!

Delightful title, and a nice feelsy little story to go with it. :pinkiehappy:

I reviewed this story as part of Read It Later Reviews #89!

My review can be found here.

CCC

10225496

A very fair and straightforward review. Thank you for it.

I have no idea what my previous comment meant about a Distraction wanting something...

But now on more fine re-read of my re-read... I wonder what happened last week...

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.

Found this through a search that went badly. Forgot I ever read it, so I'm reading it again! ... *read* ... :heart: :fluttercry::heart:

----

Typos:

Confrontation

:

one of the instinct

one of the instincts

How come did Rarity turn up at Sugarcube Corner just then?

Is he still feeling a bit fuzzy? Otherwise:

Epilogue

:

How come Rarity turned up at Sugarcube Corner just then?

CCC

10632447

one of the instincts

Fixed, thanks.

How come Rarity turned up at Sugarcube Corner just then?

...I'm pretty sure that both formulations of this sentence are grammatically correct.

10633629

I've never heard of "How come did" until, but if you're sure... :twilightsmile:

Login or register to comment