Bolyai–Gerwien theorem

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 generalized 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"; the pieces can, in theory, be cut with scissors from paper and reassembled by hand.

Higher dimensions

The analogous statement about polyhedra in three dimensions, known as Hilbert's third problem, is false, as proven by Max Dehn in 1900.

History

Farkas Bolyai, father of Janos Bolyai, first formulated the question. Gerwien proved the theorem in 1833, but in fact Wallace had proven the same result already in 1807.

According to other sources, Farkas Bolyai and Gerwien had independently proved the theorem in 1833 and 1835, respectively.

External links

* [http://www.cut-the-knot.org/do_you_know/Bolyai.shtml Wallace-Bolyai-Gerwien Theorem]
* [http://demonstrations.wolfram.com/AnExampleOfTheBolyaiGerwienTheorem/ An Example of the Bolyai-Gerwien Theorem] by Sándor Kabai, Ferenc Holló Szabó, and Lajos Szilassi, The Wolfram Demonstrations Project.


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