In the 1960s, Jim Stasheff, in order to formulate a homotopy coherent associativity structure underlying based loop spaces, discovered a fundamental object of mathematics: the associahedra. These are convex polytopes in Euclidean spaces which neatly organize higher associativity by their facets. In this talk, we first review Stasheff¹s construction, assuming a little acquaintance of the fundamental group of a topological space. Then we shall prove that the linearized associahedra admit a cyclically invariant diagonal which is unique up to homotopy. This leads to interesting applications, both in algebra and in topology.