An Example of the Bolyai-Gerwien Theorem
This conceptual design can be used to demonstrate the Bolyai–Gerwien theorem in the case of triangles and rectangles. A triangle is cut into four pieces that can be rotated to form a rectangle. (Henry Dudeney published this as a plane rearrangement puzzle in 1902.)
The Bolyai–Gerwien Theorem says that a polygon can be cut into a finite number of triangles that can be reassembled to form any other polygon having the same area. In other words: if two polygons have the same area, then they are equidecomposable (scissor-equivalent). 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.