The associahedron is the 
-dimensional
generalization of the pentagon. It was discovered by
Stasheff in 1963 and it is also known as the Stasheff polytope. The number of nodes
in the 
-associahedron
is equivalent to the number of binary trees with 
nodes, which is the Catalan
number 
.
The associahedron is the basic tool in the study of homotopy associative Hopf spaces.
Loday (2004) provides the following method for associahedron construction. Take 
, the set of planar binary trees with
leaves. Define 
as the number of leaves to the left of the 
th vertex and 
as the number of leaves to the right of the 
th vertex. For 
in 
, define
 |
The 
-associahedron
is then defined as the convex hull of 
.
The associahedron can be obtained by removing facets from the permutohedron, and is related to the cyclohedron and permutohedron.
-Cubes Operads."