Tarski's circle-squaring problem asks whether a planar disk can be partitioned into finitely many sets and reassembled
by isometries to form a square
of the same area (Laczkovich 1990). Laczkovich (1990) proved
that this is possible even using only translations.
Marks and Unger (2017) obtained such a partition with Borel sets as the pieces. The pieces need not be connected or obtainable by ordinary
geometric cuts. The resulting disk and square
are equidecomposable. This differs from circle
squaring by compass and straightedge,
which is impossible.
