The Banach-Tarski paradox states that a solid ball in has a finite set
partition into six pairwise disjointsets
of points which can be reassembled to form two disjointballs of the same size as the original (Banach and Tarski 1924).
The reassembly uses only rigid motions, specifically
rotations and translations,
without scaling. The number of pieces was subsequently
reduced to five by Robinson (1947), although the pieces are extremely complicated.
Five pieces are minimal, although four pieces are sufficient as long as the single
point at the center is neglected.
The proof uses the axiom of choice, and the pieces cannot all be measurable sets with respect to Lebesgue measure. If they were, finite
additivity and invariance under rigid motions
would force the two reassembled balls to have the same total
volume as the original ball,
which is impossible. The result is therefore not a physical cutting procedure and
does not contradict the usual rules for volume (Wagon
1993b).
has a finite set
partition into six pairwise disjoint sets
of points which can be reassembled to form two disjoint balls of the same size as the original (Banach and Tarski 1924).
The reassembly uses only rigid motions, specifically
rotations and translations,
without scaling. The number of pieces was subsequently
reduced to five by Robinson (1947), although the pieces are extremely complicated.
Five pieces are minimal, although four pieces are sufficient as long as the single
point at the center is neglected.
with nonempty interiors are equidecomposable
by rigid motions (Wagon 1993b).
