Equidissection

In geometry, an equidissection is a partition of a polygon into triangles of equal area. The study of equidissections began in the late 1960s with Monsky's theorem, which states that a square cannot be equidissected into an odd number of triangles. In fact, most polygons cannot be equidissected at all.

Much of the literature is aimed at generalizing Monsky's theorem to broader classes of polygons. The general question is: Which polygons can be equidissected into how many pieces? Particular attention has been given to trapezoids, kites, regular polygons, centrally symmetric polygons, polyominos, and hypercubes.

Equidissections do not have many direct applications. They are considered interesting because the results are counterintuitive at first, and for a geometry problem with such a simple definition, the theory requires some surprisingly sophisticated algebraic tools. Many of the results rely upon extending p-adic valuations to the real numbers and extending Sperner's lemma to more general colored graphs.

A dissection of a polygon P is a finite set of triangles that do not overlap and whose union is all of P. A dissection into n triangles is called an n-dissection, and it is classified as an even dissection or an odd dissection according to whether n is even or odd.

An equidissection is a dissection in which every triangle has the same area. For a polygon P, the set of all n for which an n-equidissection of P exists is called the spectrum of P and denoted S(P). A general theoretical goal is to compute the spectrum of a given polygon.

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