# Bolyai-Gerwien theorem

###### Theorem 1 (Bolyai-Gerwien Theorem).

To cast the theorem in more mathematically rigorous terms, let $P$ be a polygon in a plane (Euclidean plane  ). A dissection or polygonal decomposition of $P$ is a finite set  of polygons $P_{1},\ldots,P_{n}$ such that $P=P_{1}\cup\cdots\cup P_{n}$ and $P_{i}\cap P_{j}$ is either a line segment  , a point, or the empty set  . Two polygons $P$ and $Q$ are said to be equidecomposable (or scissor-equivalent) if there is a dissection $\{P_{i}\mid i=1,\ldots,n\}$ of $P$ and a finite set of rigid motions  $\{m_{i}\mid i=1,\ldots,n\}$ such that $\{m_{i}(P_{i})\mid i=1,\ldots,n\}$ is a dissection of $Q$. From this definition, it is easy to see that equidecomposability is an equivalence relation  on the set of all polygons in a given plane. Furthermore, if two polygons are equidecomposable, they have the same area, and the converse  of which is the statement of Bolyai-Gerwien Theorem.

Remarks. Farkas Bolyai conjectured this in the 1790s and William Wallace proved it in 1808. Unaware of this, Paul Gerwien proved it again in 1833, and then Bolyai, unaware of both earlier results, gave another proof in 1835.

David Hilbert wondered if the same could be done for any pair of polyhedra of equal volume (see the third of Hilbert’s problems). Dehn proved that this could not be done shortly after (Dehn’s Theorem). However, under some suitable conditions, Hadwiger proved that equidecomposability is still possible.

Title Bolyai-Gerwien theorem BolyaiGerwienTheorem 2013-03-22 16:32:19 2013-03-22 16:32:19 CWoo (3771) CWoo (3771) 11 CWoo (3771) Theorem msc 28A12 Wallace-Bolyai-Gerwien theorem Bolyai-Gerwin theorem polygonal decomposition dissection equidecomposable