Associahedron

Example

Let $n \ge 1$ and let $Q$ be a convex polygon with $n + 3$ vertices. A diagonal of $Q$ joins two non-adjacent vertices. The associahedron $K$ of dimension $n$ is a convex polytope whose face lattice is described as follows.

1. The nonempty faces of $K$ correspond to the sets of pairwise non-crossing diagonals of $Q$, ordered by reverse inclusion. A $k$-face corresponds to $n - k$ diagonals.
2. The vertices correspond to the triangulations of $Q$, the maximal sets of $n$ non-crossing diagonals.
3. The facets correspond to the single diagonals of $Q$.
4. Two vertices are joined by an edge when the triangulations differ by a flip. A flip removes one diagonal and inserts the other diagonal of the quadrilateral that appears.

Each triangulation has $n$ diagonals and each can be flipped, so every vertex lies in exactly $n$ edges, and $K$ is a simple polytope. The number of vertices is the Catalan number $C_{n+1}$, the number of triangulations of an $(n+3)$-gon.