Equidecomposable

Equidecomposable is a term used in geometry to describe two or more shapes that can be cut into a finite number of congruent pieces, which can then be rearranged to form each other. This means that if two shapes are equidecomposable, they have the same area. The concept is closely related to Wallace's Theorem and the Bolyai-Gerwien Theorem, which state that any two polygons with equal area are equidecomposable.

The process of demonstrating equidecomposability involves finding a specific set of congruent pieces that can be dissected from one polygon and reassembled to form another. This is not always trivial and can involve complex cuts and rearrangements. The pieces themselves do not have to be polygons; they can be any shape.

While the Bolyai-Gerwien Theorem applies specifically to polygons, the concept of equidecomposability can be extended to other types of shapes, though it becomes more complex. For example, it has been shown that a circle and a square of equal area are not equidecomposable in the plane. This highlights the limitations of the theorem and the nuances of geometric dissection. The study of equidecomposability has connections to topics in recreational mathematics and geometric measure theory.