M Theory Lesson 282

Recall that the vertices of associahedra are described by all rooted binary trees of a certain height, such that degeneracies in the level of the nodes is permitted. For example, for trees with three vertices (including a root) there must be four leaves, and we obtain the five vertices of a pentagon. The edges of the pentagon are labelled by trees with only two nodes, which are the contractions of the trees on the boundary vertices. And the face of the pentagon itself is labelled by the single vertex four leaved tree. The associahedron for two leaves is a point and the associahedron for three leaves is a single edge. For each real dimension, there is an associahedron.

What about ternary trees? First observe that the real dimension must increase by two at each step, because ternary vertices increase the number of leaves by two at each branching. The first two ternary polytopes are described by the following trees. The second case has three points on a surface, with no marked edges, just like a Riemann sphere. The next case naturally lives in dimension four, so we only draw the seven leaved trees marking the $12$ points: