This is the flip graph of a convex hexagon. It is planar, but I wanted to communicate (1) that it is the skeleton of a polyhedron (the associahedron), and (2) that the top and bottom vertices are at distance four from each other. I think the drawing below communicates that better than a planar drawing would have. It's a bit distorted from the projection of a symmetric convex polyhedron, I think, mainly because I wanted to avoid having the middle row of triangulations overlap each other.