Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original?
What is the true nature of this paradox ? I don't really understand this ?
2025-01-13 00:00:35.1736726435
The Banach–Tarski Paradox
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In the Banach-Tarski Paradox construction, one takes a ball in $\mathbb{R}^3$ (that is, three dimensional space, as we conceive of it in mathematics), cuts it into five pieces, then rotates and translates those pieces. In the end, one gets two balls of the same size as the original. The paradox is that you would expect these operations to preserve the total volume. The reason that they don't is because the pieces that you chop the ball up into are extremely jagged. In fact they are so jagged that there is no "reasonable" way to assign them a volume. Mathematically we say that they are not Lebesgue measurable. If they were measurable, the rotation and translation operations would preserve the volume, and the paradox could not occur.
This has no real physical significance, because it would be impossible to make the initial decomposition with any real object. The "microscale jaggedness" would require you to cut electrons in half.