Tarski problem

on "squaring the circle"

This problem, posed by A. Tarski in 1925, asks whether it is possible to partition the disc in the plane into finitely many sets which can be rearranged using isometries of the plane to form a partition of a square. A positive result has been recently (1989) announced by M. Laczkovich.

The three-dimensional analogue of this problem is easier to handle (because the group of isometries is richer). In fact, one has the Banach–Tarski paradox: If $A$ and $B$ are any two bounded subsets of $\mathbf{R}^3$ with non-empty interiors, then $A$ and $B$ are equi-decomposable. Here, two sets $A$ and $B$ are said to be equi-decomposable (with respect to the group of isometries of $\mathbf{R}^3$) if for some $n$ there are partitions $A = \cup_{i=1}^n A_i$, $B = \cup_{i=1}^n B_i$, with $A_i \cap A_j = \emptyset = B_i \cap B_j$ for $i \ne j$ and there are isometries (motions) $g_1,\ldots,g_n$ of $\mathbf{R}^3$ such that $g_i(A_i) = B_i$. The proof uses the axiom of choice. A precursor of the Banach–Tarski paradox was a paradoxical example of F. Hausdorff, and the result is also known as the Hausdorff–Banach–Tarski theorem.

In the plane the group of isometries is solvable and equi-decomposable sets must have the same measure. Two polygons in the plane are congruent by dissection if one of them can be decomposed into finitely many polygonal pieces that can be rearranged using isometries and ignoring boundaries to form the other polygon. The Bolyai–Gerwien theorem states that two polygons of equal area are congruent by dissection. Using this, Tarski showed in 1924 that two equal-area polygons in the plane are equi-decomposable, and this led to the formulation of the Tarski problem.

Cf. also Equal content and equal shape, figures of.

References

[a1]	S. Wagon, "The Banach–Tarski paradox" , Cambridge Univ. Press (1985)
[a2]	S. Banach, A. Tarski, "Sur la décomposition des ensembles de points en parties respectivement congruents" Fund. Math. , 6 (1924) pp. 244–277

References

[b1]	M. Laczkovich, "Equidecomposability and discrepancy; a solution of Tarski’s circle- squaring problem." J. Reine Angew. Math. 404 (1990) 77-117 DOI 10.1515/crll.1990.404.77 Zbl 0748.51017

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