Wallace–Bolyai–Gerwien theorem

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Wallace–Bolyai–Gerwien theorem

Summary

The Wallace–Bolyai–Gerwien theorem states that any two flat, two-dimensional polygons of equal area are equidecomposable (or scissors-congruent) — meaning one can be cut into a finite number of polygonal pieces and reassembled using only translations and rotations to form the other. 

Proved independently by William Wallace (1807), Farkas Bolyai (1833), and Paul Gerwien (1835), the theorem offers a constructive proof that does not rely on the Axiom of Choice, making the rearrangement physically achievable by slicing and rejoining the pieces (such as turning a square into an equilateral triangle of the same area). While this holds true for 2D shapes in Euclidean, hyperbolic, and spherical geometries, it notably fails in three dimensions, as shown by Max Dehn's resolution of Hilbert's third problem.

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