Wallace–Bolyai–Gerwien theorem

By the Wallace–Bolyai–Gerwien theorem, a square can be cut into parts and rearranged into a triangle of equal area.

In geometry, the Wallace–Bolyai–Gerwien theorem,[1] named after William Wallace, Farkas Bolyai and Paul Gerwien, states that any two simple polygons of equal area are equidecomposable; i.e. one can cut the first into finitely many polygonal pieces and rearrange the pieces to obtain the second polygon.

"Rearrangement" means that one may apply a translation and a rotation to every polygonal piece.

Unlike the generalized solution to Tarski's circle-squaring problem, the axiom of choice is not required for the proof, and the decomposition and reassembly can actually be carried out "physically"; the pieces can, in theory, be cut with scissors from paper and reassembled by hand.

The theorem can be understood in two steps. First, every polygon can be cut into triangles: for convex polygons this is immediate, by cutting off each vertex in turn, while for concave polygons this requires more care. Each of these triangles can then be converted to a right triangle, by dropping an altitude (that is, drawing a line perpendicular to the triangle's base and through the top vertex). This is sufficient to easily compute the area, as each right triangle is half a rectangle, or alternatively can be cut halfway up to be reassembled into a rectangle. The second and subtler step is that each right triangle (or equivalently rectangle) can be decomposed into a rectangle with a side of a given (unit) length. Once this is proven, it follows that every polygon can be decomposed into a rectangle with unit width and height equal to its area, which proves the theorem.

Higher dimensions

The analogous statement about polyhedra in three dimensions, known as Hilbert's third problem, is false, as proven by Max Dehn in 1900.

History

Farkas Bolyai first formulated the question. Gerwien proved the theorem in 1833, but in fact Wallace had proven the same result already in 1807.

According to other sources, Bolyai and Gerwien had independently proved the theorem in 1833 and 1835, respectively.

References

  1. ↑ Proceedings of the American Mathematical Society - Vol. 94, No. 2, Jun., 1985

External links