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.

In three dimensions, every stacked polytope is a polyhedron with triangular faces, and several of the deltahedra (polyhedra with equilateral triangle faces) are stacked polytopes In a stacked polytope, each newly added simplex is only allowed to touch one of the facets of the previous ones.

Thus, for instance, the quadaugmented tetrahedron, a shape formed by gluing together five regular tetrahedra around a common line segment is a stacked polytope (it has a small gap between the first and last tetrahedron).

However, the similar-looking pentagonal bipyramid is not a stacked polytope, because if it is formed by gluing tetrahedra together, the last tetrahedron will be glued to two triangular faces of previous tetrahedra instead of only one.

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.