Associahedron

The n-dimensional associahedron or Stasheff polytope is a convex n-polytope where each vertex corresponds to a binary tree of n + 1 leaves, equivalently the ways to bracket (or associate) 1 &bull; 2 &bull; ... &bull; (n + 1) where &bull; is an infix binary operator. Two vertices are connected by an edge if their bracketings differ by only one bracket. Despite the -hedron suffix, only the 3D associahedron is a polyhedron, so a more accurate name would have been "associatope."

Associahedra were originally conceived as abstract polytopes, but they have many possible realizations as convex polytopes in Euclidean space. Jean-Louis Loday devised a particularly elegant one.

The 3D associahedron can be realized as a near-miss Johnson solid with 3 squares and 6 irregular pentagons as faces. It is the result of taking a triangular bipyramid and truncating the three vertices where four triangles meet, and is the dual of the triaugmented triangular prism.

Associahedra are closely related to the permutohedra.