Equivalent and Equidecomposable Figures

Equivalent figures are plane or three-dimensional figures of identical area or volume. Two figures are equidecomposable if they can be subdivided into an equal number of pairwise congruent parts.

The concept of equidecomposability is usually applied only to polygons and polyhedrons. Equidecomposable figures are equivalent. The Hungarian mathematician J. Bolyai, in 1832, and the German mathematician P. Gerwin, in 1833, proved that equivalent polygons are equidecomposable, a result known as the Bolyai-Gerwin theorem. Any polygon can therefore be made into a square by dividing it into parts, which are then rearranged. The concept of equidecomposability is the basis of the decomposition method used to calculate the areas of polygons. Thus, a parallelogram can be reduced to a rectangle by subdivision and rearrangement, a triangle can be reduced to a parallelogram, and a trapezoid can be reduced to a triangle. The concept of equidecomposability is equivalent to that of equicomplementability, which is the basis of the completion method, wherein congruent parts are added to two figures in such a way that the resulting figures are congruent.

Equivalent polyhedrons are not always equidecomposable. For this reason, the method of exhaustion or some other method of disguised integration, for example, Cavalieri’s principle, is used to derive the formula for the volume of a triangular pyramid. According to Dehn’s theorem, a cube and a regular tetrahedron of equal volume are not equidecomposable. This theorem was proved by the German mathematician M. Dehn in 1901 and provides a negative answer to Hilbert’s third problem. In his proof, Dehn constructed a system of additive invariants that must be equal if the polyhedrons are to be equidecomposable. He showed that some of the invariants take on different values for the cube and for the regular tetrahedron of equal volume. These results were extended by the Swiss mathematician H. Hadwiger and his students. In particular, J. P. Siedler showed that the equality of the Dehn invariants of two polyhedrons is not only necessary but also sufficient for the polyhedrons to be equidecomposable.