AUTHOR NARRATION
Hear the poem.
Read by A.F. King in the original English.
Banach–TarskiA.F. King · English narration
POEM + PARADOX
Banach–Tarski
Humpty Dumpty discovers, several centuries too late, that set theory had found a way to make reassembly more astonishing than repair.
CONCEPTUAL VISUALIZATION
One sphere, two spheres.
This deliberately simplified animation communicates the conceptual shock of the paradox. It is not the actual decomposition and should not be presented as a proof.
Conceptual only: the genuine theorem uses non-measurable sets, not five ordinary wedges that could be cut with a knife.
THE MATHEMATICIANS
Banach and Tarski.
MEDIA PLACEHOLDERStefan Banach
/media/banach-tarski/stefan-banach.jpgStefan Banach
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MEDIA PLACEHOLDERAlfred Tarski
/media/banach-tarski/alfred-tarski.jpgAlfred Tarski
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NURSERY RHYME MEETS SET THEORY
The king’s men were working in the wrong century.
MEDIA PLACEHOLDERHumpty Dumpty
/media/banach-tarski/humpty-dumpty.jpgHumpty Dumpty
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Add a compact treatment of how the nursery rhyme supplies the poem’s body, fall, failed reconstruction, and comic outrage at discovering that abstract mathematics permits something far stranger than repair.
THE ACTUAL MATHEMATICS
Why does such a decomposition exist?
This section is reserved for a rigorous proof, proof sketch, or carefully selected external source. The local PDF path is already wired below.
LOCAL PROOF · PDF
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External source
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READER DISCUSSION
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