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  "body": "\n## Example\n\nLet $n \\ge 1$ and let $Q$ be a convex polygon with $n + 3$ vertices.\nA diagonal of $Q$ joins two non-adjacent vertices.\nThe *associahedron* $K$ of dimension $n$ is a [[convex polytope|300K6CB1KH]] whose face lattice is described as follows.\n\n1. 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.\n2. The [[vertices|R7V3S9V891]] correspond to the [[triangulations|Q6KKBP3RRP]] of $Q$, the maximal sets of $n$ non-crossing diagonals.\n3. The [[facets|DPDXN2FJC4]] correspond to the single diagonals of $Q$.\n4. 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.\n\nEach triangulation has $n$ diagonals and each can be flipped, so every vertex lies in exactly $n$ edges, and $K$ is a [[simple polytope|7E6FXTGH2Y]].\nThe number of vertices is the [[Catalan number|0YWZJEABB2]] $C_{n+1}$, the number of triangulations of an $(n+3)$-gon.\n\n## Remarks\n\nFor $n = 2$, $Q$ is a pentagon with $5$ triangulations and $5$ diagonals, and $K$ is a pentagon.\nFor $n = 3$, $Q$ is a hexagon with $14$ triangulations and $9$ diagonals, and $K$ is a three-dimensional polytope with $14$ vertices and $9$ facets ($3$ squares and $6$ pentagons).\nThe same lattice also describes the ways to put brackets in a product of $n + 2$ factors, which is where the name comes from.\nSeveral explicit realizations by coordinates are known.\n\n## Explanations\n\n### L1 human\n\nDraw a hexagon and cut it into triangles with straight cuts between corners.\nThere are $14$ ways to do it.\nChanging one cut at a time moves you between them, and these moves form the edges of a solid with $14$ corners.\nThat solid is the associahedron; in general its corners count the triangulations of a polygon.\n",
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