Did Futurama get the Banach-Tarski Paradox right?

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The Banach-Tarski Dupla-Shrinker recently made an appearance on an episode of Futurama. Using it, Bender got to make two, slightly smaller, copies of himself. The smaller Benders are why they added the 'shrinker' part of the name. The actual Banach-Tarski theorem allows an object to spawn a perfect copy of itself, at its exact size, just by being chopped into bits.

The paradox was first described in 1924 by Stefan Banach and Alfred Tarski. They showed that, if someone were to chop up a solid ball of any size into six very precisely-shaped pieces, those pieces could be rearranged and used to form two new solid balls, each exactly the same size as the original. The pieces would be re-positioned, but not stretched or blown up to larger proportions. Later, another mathematician reduced the number of pieces necessary to do this to five. It gets even stranger. Another version of the theory has a ball the size of a pea being chopped up and reconstructed to form a ball the size of the sun.

This seems intuitively impossible, but as an added bonus, it's theoretically impossible as well. Physics says that mass cannot be created or destroyed with nothing more than a pair of scissors. Anyone who managed to achieve a practical Banach-Tarski operation would essentially be doubling the mass of something, like magic.


It's not magic that drives this paradox. Rather, it's a certain set of assumptions and some tricks with volume. The theoretical Banach-Tarski ball can be cut into fragments that are infinitely small and thin. These fragments are special shapes for a reason. If we wanted to find the volume of a cube, we could measure one side and cube that measurement. If we encountered a rectangular solid, we could simply multiply the base area times the height. But some shapes don't have such easily defined volumes. The shapes that the ball would have to be cut into would be so jagged and scalloped that they would be more like a scatter of points than a solid. As such, their volume is nebulous enough to double if they're put in the right place. At least theoretically.

Via Wolfram Mathworld and Math Fun Facts.