Equidissection 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
Regular polytope the vertex figure. Unlike the case for Euclidean polytopes, an abstract polytope with regular facets and vertex figures "may or may not" be regular itself – for example, the square pyramid, all of whose facets and vertex figures are regular abstract polygons. The classical vertex figure will, however, be a realisation of the abstract one. The traditional way to construct a regular polygon, or indeed any other figure on the plane, is by compass and straightedge. Constructing some regular polygons in this way is very simple (the easiest is perhaps the equilateral triangle), some are more complex, and some are