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A comment on The Point of the Banach-Tarski Theorem. Parent comment

Re:The Point of the Banach-Tarski Theorem by zokier on Sunday January 22, 2023 @06:07PM (#34483322)
> It relies on uncountably-infinite division of an object

But the theorem claims finite division and not infinite?

In R³, given a solid ball B of radius R it is possible to partition B into finitely many pieces such that those pieces can be reassembled to form two solid balls B1 and B2 each of radius R

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Re:The Point of the Banach-Tarski Theorem by moloch-hai on Sunday January 22, 2023 @06:16PM (#34483403)
Finitely many pieces, but on infinitely variable boundaries.
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Re:The Point of the Banach-Tarski Theorem by wnkrshm on Monday January 23, 2023 @06:37AM (#34487676)
It's still astounding to me that the 'fractal trick' starts working in R3 and not already in R2.
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Re:The Point of the Banach-Tarski Theorem by moloch-hai on Monday January 23, 2023 @01:42PM (#34493057)
Agreed, each added dimension adds wacky surprises.

Packing spheres into a cube leaves you a pretty comprehensible space in the middle. Pack hyperspheres into a hypercube, and all hell breaks loose.

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Re:The Point of the Banach-Tarski Theorem by drdec on Sunday January 22, 2023 @11:56PM (#34485641)
Keep in mind that "pieces" might not mean the same thing you think of it as meaning.
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