Wallace–Bolyai–Gerwien theorem

Any two polygons of equal area can be partitioned into finitely many polygonal pieces and rearranged by rigid motions to form one another. It characterizes equal-area polygons by finite equidecomposability.

Connect

Polygon triangulation: The partition of a polygon into triangles using noncrossing diagonals between its vertices. Triangulating both polygons reduces the problem to rearranging triangles.
Farkas Bolyai: A Hungarian mathematician who studied geometry and published work on polygonal equivalence. His work helped establish the dissection result associated with the theorem.
Polygon: A plane figure bounded by a finite closed chain of straight line segments. The theorem applies specifically to polygons, not arbitrary plane regions.
Hilbert's third problem: The problem of whether equal-volume polyhedra can always be dissected into congruent pieces. It asks for a three-dimensional analogue that fails, unlike the planar theorem.
Triangle dissection: A finite partition of one triangle into polygonal pieces that can be reassembled into another shape. The central geometric step converts equal-area triangles into matching pieces.
Pál Gerwien: A German mathematician who published an early proof that equal-area polygons are equidecomposable. His proof is one source of the theorem’s name.
Rigid motion: A transformation of the plane that preserves distances and angles. Rearrangement permits moving and turning pieces without deforming them.
Dehn invariant: An invariant of a polyhedron, built from its edge lengths and dihedral angles, that obstructs scissors congruence. It supplies an obstruction absent from the equal-area classification of polygons.
Congruence (geometry): A relation between geometric figures that can be mapped onto one another by a distance-preserving transformation. Each piece must move without changing its shape or size.
William Wallace: A Scottish mathematician who published a proof of polygonal equivalence by dissection. His independent result supplies the theorem’s first eponym.