Associahedron

Definition
The associahedron or Stasheff polytope with parameter $$n$$, sometimes denotes as $$K_n$$, is a polytope whose vertices correspond to all the ways of parenthesizing a product string of length $$n$$, and where the faces are defined as follows:


 * Two vertices are part of an edge (i.e., a face with two vertices) if and only if a single application of the associativity law gets from one vertex to the other.

The associahedron $$K_4$$ is particularly important for understanding associative and non-associative operations and it is termed the associativity pentagon.