M Theory Lesson 184

Recall that single chorded polygons label the faces (codimension 1 objects) of an associahedron. Codimension 2 objects are labelled by non-intersecting two chorded polygons. For example, the two dimensional polytope has 5 faces labelled by the 5 chords of a pentagon. The 5 non-crossing two chord diagrams give the 5 vertices. By including the crossed chord diagrams, one effectively describes (the dual of) a full simplex (in 4D) with 10 vertices and 5 faces. For any chorded polygon, the choice of an arbitrary pair of chords amounts to the choice of an arbitrary pair of faces on the polytope. If one represents faces by points, two chords represent an edge joining two points, and one always obtains a full higher dimensional n-simplex $K_{n+1}$. The (dual) associahedra then appear as subgraphs of the complete graph $K_{m}$ for $m$ in this sequence.