Equidissection

The decomposition into triangles of equal area (also direct fragmentation ) is a problem of plane geometry. This evaluation will, among other things, whether the decomposition of a given polygon into equal-area triangles is at all possible.

The research on this problem began in the late 1960s with Paul Monskys theorem, according to which a square can not be decomposed into an odd number of triangles of equal area. The proof uses valuation theory and is the only known proof of this theorem.

In fact, most polygons can not be divided into triangles of equal area. This raises the question: Which polygons can be decomposed into as many parts of equal area? Were examined in particular harnesses, kite quadrilaterals, regular polygons, point-symmetrical polygons and polyominoes and the decomposition of hyper-cubes in simplices. In the case of regular, n -vertex polygons with n ≥ 5 showed Elaine Kasimatis that they can be decomposed into m equal -area triangles only if m is a multiple of n. For n = 3 or n = 4, this is obviously not correct: A square can be decomposed into two equal -area triangles and a triangle into any number.

Decompositions in equal-area triangles have few direct applications. However, they are considered interesting because the results at first glance often contradict the expectations and the theory of a geometric problem with such a simple definition requires surprisingly sophisticated algebraic tools. Many results are based on the application of valuation theory to the real numbers, and the coloring in graph theory on the basis of the lemma of Sperner.