Associahedron

In mathematics, an associahedron K_n is an (n – 2)-dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of n letters, and the edges correspond to single application of the associativity rule. Equivalently, the vertices of an associahedron correspond to the triangulations of a regular polygon with n + 1 sides and the edges correspond to edge flips in which a single diagonal is removed from a triangulation and replaced by a different diagonal. Associahedra are also called Stasheff polytopes after the work of Jim Stasheff, who rediscovered them in the early 1960s after earlier work on them by Dov Tamari. The one-dimensional associahedron K3 represents the two parenthesizations ((xy)z) and (x(yz)) of three symbols, or the two triangulations of a square. It is itself a line segment. The two-dimensional associahedron K4 represents the five parenthesizations of four symbols, or the five triangulations of a regular pentagon. It is itself a pentagon and is related to the pentagon diagram of a . The three-dimensional associahedron K5 is an enneahedron with nine faces (three disjoint quadrilaterals and six pentagons) and fourteen vertices, and its dual is the triaugmented triangular prism. Initially Jim Stasheff considered these objects as curvilinear polytopes. Subsequently, they were given coordinates as convex polytopes in several different ways; see the introduction of for a survey. One method of realizing the associahedron is as the secondary polytope of a regular polygon. In this construction, each triangulation of a regular polygon with n + 1 sides corresponds to a point in (n + 1)-dimensional Euclidean space, whose ith coordinate is the total area of the triangles incident to the ith vertex of the polygon. For instance, the two triangulations of the unit square give rise in this way to two four-dimensional points with coordinates (1, 1/2, 1, 1/2) and (1/2, 1, 1/2, 1). The convex hull of these two points is the realization of the associahedron K3.