Associahedron
[2] The one-dimensional associahedron K3 represents the two parenthesizations ((xy)z) and (x(yz)) of three symbols, or the two triangulations of a square.
The three-dimensional associahedron K5 is an enneahedron with nine faces (three disjoint quadrilaterals and six pentagons) and fourteen vertices, and its dual is the triaugmented triangular prism.
Subsequently, they were given coordinates as convex polytopes in several different ways; see the introduction of Ceballos, Santos & Ziegler (2015) for a survey.
For instance, the two triangulations of the unit square give rise in this way to two four-dimensional points with coordinates (1, 1/2, 1, 1/2) and (1/2, 1, 1/2, 1).
Another realization, due to Jean-Louis Loday, is based on the correspondence of the vertices of the associahedron with n-leaf rooted binary trees, and directly produces integer coordinates in (n − 2)-dimensional space.
The ith coordinate of Loday's realization is aibi, where ai is the number of leaf descendants of the left child of the ith internal node of the tree (in left-to-right order) and bi is the number of leaf descendants of the right child.
[4] It is possible to realize the associahedron directly in (n − 2)-dimensional space as a polytope for which all of the face normal vectors have coordinates that are 0, +1, or −1.
Each vertex in the above Hasse diagram has the ovals from the 3 adjacent faces. Faces whose ovals intersect do not touch.
