Banach–Tarski Paradox

Takeaway

With the axiom of choice, a 3D ball can be partitioned into finitely many non-measurable pieces and reassembled into two balls identical to the original.

The problem (before → after)

Before: Volume seems additive and preserved by rigid motions.
After: Choice enables paradoxical decompositions under group actions, revealing limits of measure and our geometric intuition.

Mental model first

It’s like cutting a “snowflake” made of infinitely detailed dust into a few clouds and rearranging them; the dust is so wild that “volume” isn’t defined for the pieces, so conservation of volume is not violated.

Just-in-time concepts

Axiom of choice; non-measurable sets.
Free groups in rotations of the sphere; paradoxical decompositions.
Amenability: Groups lacking paradoxical decompositions (e.g., abelian) vs non-amenable ones (contain free groups).

First-pass solution

Use choice to pick one representative from each orbit under a free subgroup of rotations; assemble pieces per group partitions; translate back to balls via isometry.

Banach–Tarski holds in ≥3D but not in 2D under isometries.
Amenability blocks paradoxical decompositions; additivity of measure fails for non-measurable sets.
Strength depends on allowed motions; with measurable pieces, no paradox.

Principles, not prescriptions

Set-theoretic choices can defy geometric intuition; measurability matters.
Group actions determine possible decompositions.

Common pitfalls

Thinking volume is created; the pieces lack well-defined volume.
Assuming paradox implies physical possibility; it’s purely mathematical.

Connections and contrasts

See also: [/blog/axiom-of-choice], [/blog/cantors-diagonal-argument], [/blog/four-color-theorem] (contrast constructive proofs).

Quick checks

Why 3D not 2D? — Rotations in 3D contain free groups; 2D is amenable.
Is volume conserved? — Not defined for the pieces; no contradiction to measure theory.
Does physics allow it? — No; real matter is atomic and pieces are nonconstructive.