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Tarski's circle-squaring problem - mklabgr - 07-18-2026
Tarski's circle-squaring problem
Summary
Tarski’s circle-squaring problem, posed by Alfred Tarski in 1925, asks whether a circle can be cut into a finite number of pieces and rearranged to form a square of exactly the same area. Unlike the ancient problem of squaring the circle with only a compass and straightedge—which is impossible—Tarski’s version allows the shapes to be cut apart and reassembled. For many years, mathematicians did not know whether such a decomposition was possible until Miklós Laczkovich proved in 1990 that it could indeed be done.
His proof, however, relied on highly abstract mathematics, the axiom of choice, and an enormous number of non-constructive pieces, making it impossible to visualize or carry out in practice. Later work by other mathematicians produced constructive solutions using Borel pieces, showing that the result can be achieved in a more explicit and mathematically satisfying way. The problem has become a landmark in geometry, illustrating how seemingly impossible puzzles can sometimes be solved when the rules are carefully redefined.
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