Stacked polytope

In polyhedral combinatorics (a branch of mathematics), a stacked polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets. For instance, the graphs of three-dimensional stacked polyhedra are exactly the Apollonian networks, the graphs formed from a triangle by repeatedly subdividing a triangular face of the graph into three smaller triangles.

One reason for the significance of stacked polytopes is that, among all d-dimensional simplicial polytopes with a given number of vertices, the stacked polytopes have the fewest possible higher-dimensional faces. For three-dimensional simplicial polyhedra the numbers of edges and two-dimensional faces are determined from the number of vertices by Euler's formula, regardless of whether the polyhedron is stacked, but this is not true in higher dimensions. Analogously, the simplicial polytopes that maximize the number of higher-dimensional faces for their number of vertices are the cyclic polytopes.