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> 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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Keep in mind that "pieces" might not mean the same thing you think of it as meaning.

Finitely many pieces, but on infinitely variable boundaries. It was clever to make the proof allow a finite number in that place. Without, it would have attracted no attention.

It's still astounding to me that the 'fractal trick' starts working in R3 and not already in R2.

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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