associahedron

thumb|Associahedron (front) thumb|Associahedron (back) thumb| is the Hasse diagram of the [[Tamari lattice .]] thumb|The 9 faces of Each vertex in the above Hasse diagram has the ovals from the 3 adjacent faces. Faces whose ovals intersect do not touch.

In mathematics, an associahedron is an -dimensional convex polytope in which each vertex corresponds to a way of correctly inserting opening and closing parentheses in a string of 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 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.