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The Banach-Tarski paradox challenges our sense of volume. The Banach-Tarski paradox challenges our sense of volume.

Cutting a ball into the impossible

By Peter Teoh, Science Writer

The Banach-Tarski paradox claims a solid ball can be split and reassembled into two identical balls. It sounds like magic, but it lives in the strange world of infinite sets.

Explainer: Why volume can break

Focus: The paradox uses the axiom of choice to pick points from sets that are not measurable. Those pieces cannot be assigned normal volume, so the usual conservation of volume fails.

The result does not work in the real world because the pieces are wildly scattered and cannot be physically constructed. It is a reminder that geometry changes when you allow infinite, non-measurable sets.

Summary of Key Ideas:

  • The paradox depends on the axiom of choice.
  • Non-measurable sets do not have a well-defined volume.
  • Physical matter cannot realize the construction.

Side Notes

  • The paradox works in 3D and higher, not in 2D.
  • It uses only rotations and translations, no stretching.
  • Infinity and measure theory explain the result.
  • How choice axioms reshape geometry.

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