openai/gpt-5-pro 📝❓🖼️🔢 → 📝

Short version: In three dimensions, you can take a solid ball, cut it into finitely many disjoint pieces, and by using only rigid motions (rotations and translations), reassemble those pieces into two solid balls each the same size as the original. This does not violate conservation of volume because the pieces are so wildly non-measurable that “volume” is undefined for them. The construction crucially uses the axiom of choice.

Key points

• What the theorem says:

  • There exists a partition of a solid ball in R^3 into finitely many sets that can be rigidly moved to form two balls congruent to the original.
  • More strongly: in R^n for n ≥ 3, any two bounded sets with nonempty interiors are equidecomposable by rigid motions.

• Why it isn’t a physical paradox:

  • The pieces are not “nice” regions. They are non-measurable sets: no volume can be assigned to them in a way that is countably additive and invariant under rigid motions.
  • Real-world cutting only permits measurable pieces; the Banach–Tarski pieces are purely set-theoretic and require infinitely precise, non-constructive choices.

• Where the axiom of choice enters:

  • The construction selects one representative from each orbit of a certain group action on the sphere, which cannot be done without some form of choice.
  • In set-theoretic universes where every set of reals is measurable (possible if you drop choice and assume large cardinals), Banach–Tarski fails.

• How the proof works (very high level):

  1. Inside the rotation group SO(3) there is a copy of the free group on two generators, F2. This group has a “paradoxical” property: it can be partitioned into parts that, after left-multiplication by group elements, cover the whole group twice.
  2. Let that free group act on the unit sphere S^2. Use the axiom of choice to pick a representative from each orbit; this yields highly non-measurable sets.
  3. Transfer the group’s paradoxical decomposition to S^2 (actually S^2 minus a countable set of exceptional points) using only rotations, so you can rotate pieces to obtain two copies of almost the whole sphere.
  4. Extend from the sphere to the solid ball by slicing along rays from the center (and bookkeeping the exceptional set). The classic construction uses 5 pieces for the ball.

• Why it only happens in dimension ≥ 3:

  • The isometry group of the plane is amenable (intuitively, it doesn’t contain a free subgroup like F2), and Tarski’s theorem says amenability rules out paradoxical decompositions. So no Banach–Tarski phenomenon in 1D or 2D with rigid motions.
  • In 2D you still can do “circle squaring” (decompose a disk into finitely many pieces and translate them to form a square of the same area), but that preserves area; it’s not paradoxical.

• What it really shows:

  • There is no finitely additive, rotation- and translation-invariant “volume” function defined on all subsets of R^3 extending Lebesgue measure. Lebesgue measure works precisely because it excludes pathological sets like those used in the decomposition.

If you’d like, I can sketch one of the concrete group-theoretic decompositions or explain the connection to amenability and Tarski’s theorem in a bit more detail.