Cut One Ball Into Two Identical Balls

A solid ball goes in on the left. A knife makes five cuts. Two whole balls, each the same size as the first, come out on the right. Nothing is stretched, nothing is shrunk, nothing is smuggled in from outside. It works because the five pieces are not solid chunks any real knife could cut - they are scattered clouds of single points that no rule for size can measure at all, and this lesson is about why that has to be true.

Before you start. All you need is the everyday idea that two solid balls can be “the same size,” and enough trust in a proof to follow it even when it fights your gut. This note stands on its own; nothing else in Mind Blown needs to come first.

The idea in 30 seconds

Banach and Tarski’s 1924 proof cuts a solid ball into five pieces and, by only turning and sliding them, reassembles them into two solid balls the same size as the original.
The proof uses the axiom of choice exactly once, to pick one point from each of an enormous scatter of point-clusters, with no rule for which point.
Volume is a rule that hands back a number for most shapes but simply has no answer for some - not zero, undefined - and at least one of the five pieces is like that.
“Volume before equals volume after” is not being broken. It stops being a meaningful sentence the moment one of your pieces has no volume to begin with.
The same trick provably cannot be done to a flat shape, and cannot be done to a real object made of a finite number of atoms.

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