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
Petrie polygon of Petrie polygons was later extended to semiregular polytopes. The Petrie polygon of the regular polyhedron {"p", "q"} has "h" sides, where The regular duals, {"p","q"} and {"q","p"}, are contained within the same projected Petrie polygon. The regular Kepler–Poinsot polyhedra have hexagonal, {6}, and decagrammic, {10/3}, Petrie polygons. Infinite regular skew polygons (apeirogon) can also be defined as being the Petrie polygons of the regular tilings, having angles of 90, 120, and 60 degrees of their square, hexagon and triangular faces respectively. Infinite regular skew polygons also exist as Petrie polygons of the regular hyperbolic tilings, like the order-7 triangular