Given a convex hexagon and a triangulation of it, each vertex can be assigned a number (shown left) given by the sum of the areas of the triangles incident to it. Together, each triangulation of the hexagon yields a coordinate in \(\mathbb{R}^6\). By the Gelfand-Kapranov-Zelevinsky theory on secondary polytopes, taking the convex hull of these coordinates results in a geometric realization of the associahedron (shown right), lying in a 3D subspace of \(\mathbb{R}^6\). As one deforms the underlying hexagon, the associahedron also deforms continuously. Notice that each facet of the associahedron corresponds to a diagonal of the hexagon (colored appropriately), and each vertex corresponds to a specific triangulation of the hexagon.

Read our paper presented at the Symposium on Computational Geometry here.

Drag vertices to deform hexagon.