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Bolyai-Gerwien theorem

In geometry, the Bolyai-Gerwien theorem states that if two simple polygons of equal area are given, one can cut the first into finitely many polygonal pieces and rearrange the pieces to obtain the second polygon.

"Rearrangement" means that one may apply a translation and a rotation to every polygonal piece.

Unlike the solution to Tarski's circle squaring problem, the axiom of choice is not required for the proof, and the decomposition and reassembly can actually be carried out "physically".

Higher dimensions

The analogous statement about polyhedra in three dimensions, known as Hilbert's third problem, is false. This was proven by Max Dehn in 1900. The answer is unknown for dimensions higher than 3.


Wolfgang Bolyai first formulated the question. Gerwien proved the theorem in 1833, but in fact William Wallace had proven the same result already in 1807.