Abstract: Planar rooted binary trees (or binary bracketings, or triangulations of a polygon) form a partially ordered set, where cover relations are given by right rotations (or right associators, or slope-increasing diagonal flips). This partially ordered set is known as the Tamari lattice T(n); its Hasse diagram is the graph of the associahedron. By erasing the last leaf (or letter, or triangle), one obtains a poset projection T(n) -> T(n-1). I will explain how a modification of these projections retrieves a broad family of alternative orientations on the graphs of associahedra, known in the literature as accordion lattices (this is joint work with Vincent Pilaud). Time permitting, I will explain how the idea extends from associahedra to other polytopes.