## Topology riddle

I'm reading "Surely you're joking, Mr. Feynman!" which is one of the funniest and interesting books I've ever read!!

However, in one paragraph, Feynman is talking with some topology students about a riddle:

The topology students: "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?"
"No holes?"
"No holes!"
"Impossible! There ain't no such a thing."
"Ha! We got him! Everybody gather around! It's So-and-so's theorem of immeasurable measure!"
Just when they think they've got me, I remind them, "But you said an orange! You can't cut the orange peel any thinner than the atoms."...

I understood Feynman argument, but I didn't get the idea of the theorem of immeasurable measure. How can you get an orange as big as the sun?

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It's related to this joke:

 Q: What's an anagram of Banach-Tarski? A: Banach-Tarski Banach-Tarski.
Google for it.

 The point of the story is that mathematicians are always trying to show that intuitive thinking will lead you astray. So they love coming up with examples of something you'd think should be impossible and then they "prove" it's true. Feynman's point it that if they really are honest about their definitions, things really do make sense if you just look at them the right way.

## Topology riddle

I was just reading that book, and was writing my Theory of Knowledge essay.
I also was wondering if the theorem of immeasurable measure existed and if so, what is it?
Feynman is correct in saying that the topologists asked him about splitting an orange finite number of times, so for an orange it was impossible, but as a theory it intrigued me as it could be a 'real' theory where you assume nothing.

 See the Banach-Tarski theorem: http://en.wikipedia.org/wiki/Banach-Tarski_paradox
 As long as we're on the topic of Feynman, in the same book was a story of how he was able to calculate cube roots using pad and pencil faster than someone using an abacus. Does anyone here know how to calculate cube roots on an abacus?
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