# Dual associahedron

True facts that are interesting (to me), relevant, true, easily verified, and yet lack the reliable sources needed to add them to the appropriate Wikipedia article, #117253:

The triaugmented triangular prism, a convex polyhedron with 14 equilateral triangle faces formed by gluing pyramids onto the square faces of a triangular prism and one of the 92 Johnson solids, has as its dual the three-dimensional associahedron, a 14-vertex polyhedron with six pentagonal faces and three quadrilateral faces that is one of the near-miss Johnson solids (it looks like it should be possible to form with all faces regular, but no).

Some more facts that I can source but haven't yet added to Wikipedia: The associahedron, or Stasheff polytope, has vertices representing the parenthesizations of a sequence of objects, and edges representing applications of the associative law to change from one parenthesization to another. The number of vertices is, in general, a Catalan number. The 14 vertices of the three-dimensional associahedron represent parenthesizations of a five-element sequence. For instance, there is an edge connecting the vertices representing the parenthesizations (a(bc))(de) and ((ab)c)(de), corresponding to the application of the associative law to the triple abc. The same polyhedron also represents the triangulations of a hexagon and flips between triangulations.