The convex hull of these two points is the realization of the associahedron "K"3. Although it lives in a 4-dimensional space, it forms a line segment (a 1-dimensional polytope) within that space. Similarly, the associahedron "K"4 can be realized in this way as a regular pentagon in five-dimensional Euclidean space, whose vertex coordinates are the cyclic permutations of the vector (1, 2 + φ, 1, 1 + φ, 1 + φ) where φ denotes the golden ratio. Because the possible triangles within a regular hexagon have areas that are integer multiples of each other, this construction can be used to give integer coordinates (in six dimensions) to the three-dimensional associahedron "K"5; however (as the example of "K"4 already shows) this construction in general leads to irrational numbers as coordinates. 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 "i"th coordinate of Loday's realization is "aibi", where "ai" is the number of leaf descendants of the left child of the "i"th internal node of the tree (in left-to-right order) and "bi" is the number of leaf descendants of the right child. 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. There are exponentially many combinatorially distinct ways of doing this. Because "K"5 is a polyhedron only with vertices in which 3 edges come together it is possible for a hydrocarbon to exist (similar to the Platonic hydrocarbons) whose chemical structure is represented by the skeleton of "K"5. This “associahedrane” C14H14 would have the SMILES notation: C12-C3-C4-C1-C5-C6-C2-C7-C3-C8-C4-C5-C6-C78. Its edges would be of approximately equal length, but the vertices of each face would not necessarily be coplanar.