Wallace–Bolyai–Gerwien theorem

It answers the question when one polygon can be formed from another by cutting it into a finite number of pieces and recomposing these by translations and rotations.

The most common version uses the concept of "equidecomposability" of polygons: two polygons are equidecomposable if they can be split into finitely many triangles that only differ by some isometry (in fact only by a combination of a translation and a rotation).

In this case the Wallace–Bolyai–Gerwien theorem states that the equivalence classes of this relation contain precisely those polygons that have the same area.

By doing this for each triangle, the polygon can be decomposed into a rectangle with unit width and height equal to its area.

In the formulation of the theorem using scissors-congruence, the use of this intermediate can be reformulated by using the fact that scissor-congruences are transitive.

Depending on the polygons, it is possible to estimate upper and lower bounds for σ(P,Q).

In two-dimensional hyperbolic and spherical geometry, the theorem holds.