The associahedron is a "mythical" polytope encoding Catalan combinatorics (for example, the vertices, counted by Catalan numbers, encode triangulations of a convex polygon). The associahedron was first "sighted" in the sixties by Stasheff and Milnor, but the first published construction is from 1989.

We report about a several very natural constructions - as fiber polytopes, via root systems, and as Minkowski sums. Surprisingly, these constructions produce distinct results, which are not affinely equivalent. But we get even more: we describe a construction that turns different triangulations into different triangulations. Thus we obtain Catalan-many realizations of the associahedron.
(Joint work with Cesar Ceballos and Francisco Santos).