AUTHOR NARRATION
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Read by the author in the original English.
Banach–TarskiEnglish narration
POEM + PARADOX
Banach–Tarski
THE POEM IN IMAGES
Humpty, reassembled.
Five illustrations follow the poem as you hear it.
“Just picture the hatching comprehension,”
FIVE PASSAGES · ONE POEM
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THE ACTUAL MATHEMATICS
The Theorem.
IN PLAIN LANGUAGE
A solid ball can be divided into finitely many sets of points, and those sets can then be moved using only rotations and translations to form two solid balls, each the same size as the original. The “pieces” are extraordinarily irregular, non-measurable sets—not objects that could be cut from physical matter—so ordinary volume does not apply to them individually.
FORMAL STATEMENT
Assuming the axiom of choice, let A and B be bounded subsets of ℝ3 with nonempty interiors. There are finite partitions A = A1 ∪ ··· ∪ An and B = B1 ∪ ··· ∪ Bn, together with Euclidean isometries g1, …, gn, such that gi(Ai) = Bi for every i. In other words, A and B are equidecomposable.
Read a complete proof (PDF) →THE MATHEMATICIANS
Banach and Tarski.
In 1924, Stefan Banach and Alfred Tarski published the theorem that now bears their names.
MEDIA PLACEHOLDERStefan Banach
/media/banach-tarski/stefan-banach.jpgStefan Banach
Polish mathematician (1892–1945), founder of modern functional analysis and a leading figure in the Lwów School of Mathematics.
MEDIA PLACEHOLDERAlfred Tarski
/media/banach-tarski/alfred-tarski.jpgAlfred Tarski
Polish-American mathematician and logician (1901–1983), renowned for foundational work on truth, logic, set theory, and model theory.
Images: Stefan Banach, public domain; Alfred Tarski, photograph by George M. Bergman, GFDL 1.2 — Wikimedia Commons.
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